\documentclass[11pt]{amsart} \usepackage[ textheight=9in, textwidth=7in, centering ]{geometry} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsbsy} \usepackage{bm} \usepackage{mathtools} \usepackage{array} \usepackage{enumitem} \usepackage{xcolor} \usepackage{booktabs} \usepackage[noadjust]{cite} \usepackage{hyperref} \usepackage[noabbrev,capitalise,nameinlink]{cleveref} \newcommand{\EC}[1]{\bgroup\color{red}#1\egroup} \hypersetup{hidelinks} \setlength{\parskip}{0em} \allowdisplaybreaks \raggedbottom \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}[theorem]{Question} \newtheorem{problem}[theorem]{Problem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{claim}{Claim}[theorem] \newtheorem*{theorem*}{Theorem} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{hypothesis}[theorem]{Hypothesis} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \crefname{hypothesis}{Hypothesis}{Hypotheses} \crefname{conjecture}{Conjecture}{Conjectures} \newcommand{\K}{\mathcal K} \newcommand{\Ltwo}{L_{2'}} \newcommand{\CC}{\mathcal C} \DeclareMathOperator{\Inv}{Inv} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\PSL}{PSL} \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\Out}{Out} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{#1}}} \newcommand{\dfn}[1]{\emph{#1}} \newcommand{\CD}[1]{\textcolor{orange}{\textsf{#1}}} \newcommand{\KO}[1]{\textcolor{green}{\textsf{#1}}} \begin{document} \title[]{The Unbalanced Group Theorem} \author{Axiom Math} \begin{abstract} The classification of finite simple groups (CFSG) identifies the basic building blocks of all finite groups. It grew from nineteenth-century work of Galois and Jordan into a collective effort involving many mathematicians, with Gorenstein playing a central organizing role. Its proof is a vast network of interdependent results such as the Feit--Thompson odd-order theorem, the signalizer functor theorem, and Aschbacher's classical involution theorem. The original papers make up the first-generation proof. The second-generation treatment, initiated by Gorenstein, Lyons, and Solomon, reorganizes and simplifies much of this work. A key structural result in the first-generation program is the Unbalanced Group Theorem (UGT). Under the inductive hypotheses of the classification, it says that a nontrivial subgroup of odd order normal in the centralizer of an involution forces the ambient simple group to be a group of Lie type in odd characteristic or one of the alternating groups of certain degrees. While formalizing the first-generation proof, we found counterexamples to a closure assertion used in a published route to UGT. This obstruction concerns that particular argument. UGT also has other proofs. Here we explain the obstruction and develop an alternative argument for the simple-group case using Solomon's 1976 theory of maximal $2$-components, which provides a framework for comparing the simple pieces arising in different involution centralizers. \end{abstract} \maketitle \section{Overview} The classification of finite simple groups (CFSG) is a monumental achievement in the history of mathematics. A group is \dfn{simple} if it is nontrivial and has no normal subgroups other than itself and the identity subgroup. Every finite group can be broken into simple factors by a composition series. Indeed, the Jordan--H\"older theorem says that the resulting factors, counted with multiplicity, do not depend on the series. Classifying these factors does not tell us every way to assemble a finite group, but it tells us which building blocks are available. The CFSG catalogs every finite simple group: \begin{theorem*} [Classification of finite simple groups] Every finite simple group is either cyclic of prime order, an alternating group $A_n$ for $n\ge5$, a simple group of Lie type, or one of the twenty-six sporadic groups, named after their discoverers apart from the Baby Monster and the Monster. \end{theorem*} The proof grew into a collective enterprise spanning much of the twentieth century. Richard Brauer made centralizers of elements of order two central to the recognition of simple groups. Walter Feit and John Thompson's odd-order theorem, published in 1963, showed that every nonabelian finite simple group has such an element \cite{FeitThompson}. Michio Suzuki and Thompson made powerful tools for recovering a group from its subgroups, while Daniel Gorenstein gave the effort a guiding program and an organizing framework. Michael Aschbacher, Richard Lyons, Ronald Solomon, and many others supplied the structural and recognition theorems that carried that program forward. Solomon's historical account \cite{SolomonHistory} describes both the ideas and the extraordinary collaboration behind them. What is now referred to as the \emph{first-generation proof} is made up of an extensive web of papers, whose main framework was already established by the early 1980s. Gorenstein's 1983 volume \cite{Gorenstein} records much of the relevant work. Work toward completion and verification continued. For example, the substantial remaining case of quasithin groups was resolved by Aschbacher and Stephen Smith in work published in 2004. Aschbacher's account \cite{AschbacherStatus} explains the distinction between the early announcements and the work needed to secure the proof. Beginning in the early 1980s, Gorenstein, Lyons, and Solomon also undertook a second-generation project, seeking to reorganize and simplify large parts of the proof into a systematic series of volumes. These efforts reflect the scale of the achievement. CFSG should be thought of as a problem of identification. Suppose there were a finite simple group of smallest order missing from the ``known list''. All the simple groups arising as sections of its proper subgroups would already be known. One then examines those subgroups, compares the information they provide, and proves that the supposedly new group must itself be one of the known groups. To this end, group theorists have made great use of centralizers of involutions: the subgroups consisting of all elements that commute with a given element of order two. Such centralizers are proper subgroups of a nonabelian simple group, but their structure can constrain the whole group. The Unbalanced Group Theorem (UGT) is one of the key results that makes this ``local-to-global'' strategy work. Roughly speaking, if even one involution centralizer has a nontrivial normal subgroup of odd order, then the ambient simple group belongs to particular familiar families. To make this precise, we recall some definitions. Throughout, all groups in this paper are finite. Let $G$ be a group. An \dfn{involution} is an element of order two. The \dfn{centralizer} of a subset $A$ of $G$ is \[ C_G(A)=\{g\in G:ga=ag\text{ for every }a\in A\}. \] We denote this centralizer by $C_G(t)$ when $A=\{t\}$. The \dfn{odd core} $O(G)=O_{2'}(G)$ is the largest normal subgroup of $G$ of odd order. It exists because the product of two normal subgroups of odd order is again a normal subgroup of odd order. The subscript $2'$ means ``prime to two.'' Normality depends on the ambient group. In particular, a subgroup can be normal in $C_G(t)$ without being normal in $G$. Although a nonabelian simple group $G$ has no nontrivial normal subgroup of odd order, an involution centralizer $C_G(t)$ may have one. The unbalanced-group problem asks what this tells us about $G$. \begin{problem}[Unbalanced-group problem: simple-group case] Which nonabelian finite simple groups $G$ contain an involution $t$ for which $O(C_G(t))\ne 1$? \end{problem} Such a simple group is called {\it unbalanced}. Generally, a finite group $G$ is unbalanced if $O(C_G(t))\nleq O(G)$ for some involution $t\in G$. For a nonabelian simple group $G$, the Feit--Thompson theorem \cite{FeitThompson} implies $G$ has even order and hence $O(G)=1$, so the two conditions agree. The Unbalanced Group Theorem answers this question under the inductive hypothesis used in the classification program, which we now make precise. A \dfn{section} of a group $X$ is a quotient $A/N$, where $A\le X$ and $N\unlhd A$. Let $\K$ be the catalog of {\it known nonabelian finite simple groups}. Namely, they are the alternating groups, the simple groups of Lie type, and the twenty-six sporadic groups. A \dfn{$\K$-group} is a group whose nonabelian simple sections belong to this catalog. A group is \dfn{$\K$-proper} if every proper subgroup is a $\K$-group. This condition is natural in a classification proof. If $G$ is a nonabelian simple group of least order outside the catalog, every nonabelian simple section of a proper subgroup has smaller order. Consequently $G$ is $\K$-proper. Groups of Lie type are the finite groups constructed from algebraic groups over finite fields, together with their twisted analogs. The groups $\PSL_n(q)$ are examples; they are the determinant-one matrices over the finite field of $q$ elements modulo the scalar matrices. Similarly, $\PGL_n(q)$ is the group of all invertible matrices modulo the nonzero scalar matrices. We say that a simple group is of \dfn{odd Lie type} if it is isomorphic to a simple group of Lie type in odd characteristic. There are small coincidences such as $A_5\cong\PSL_2(5).$ We now state the simple-group case of the Unbalanced Group Theorem. \begin{theorem}[UGT] \label{thm:UGT} Let $G$ be a nonabelian simple $\K$-proper group. If $O(C_G(t))\ne1$ for some involution $t\in G$, then either $G$ is of odd Lie type, or $G\cong A_{4a+3}$ for some positive integer $a$. \end{theorem} \begin{remark} The version of UGT stated above has been formalized in Lean in the Axiom Math CFSG repository. Our project website provides a link to the formal proof. \end{remark} \cref{thm:UGT} gives an important reduction in the classification program. Suppose that $G$ were a nonabelian simple group of smallest order outside the known catalog. Its proper subgroups satisfy the required inductive hypothesis. If $O(C_G(t))\ne1$ for some involution $t$, the theorem would identify $G$ with a group in the catalog, a contradiction. Therefore, any such counterexample must satisfy \[ O(C_G(t))=1 \qquad\text{for every involution }t\in G. \] UGT therefore removes all unbalanced groups from the remaining classification problem. The theorem also has an important connection with the \dfn{$B$-theorem}. Roughly speaking, the $B$-theorem ensures that the simple pieces visible after removing the odd core of an involution centralizer can be realized by suitable subgroups of the centralizer itself. More precisely, when $O(G)=1$, every quasisimple component of $C_G(t)/O(C_G(t))$ is the image of a quasisimple component of $C_G(t)$. We define components in \cref{sec:preliminaries}. This lifting property allows one to study actual subgroups of $G$, compare them across different centralizers, and apply recognition theorems to identify $G$. For a smallest counterexample to CFSG, the displayed consequence of UGT makes this lifting property immediate. Namely, there is no odd core to remove. The corresponding $B$-property for its proper subgroups and their sections follows from independently established results for $\K$-groups. Therefore, the simple-group form of UGT supplies the ambient reduction needed here without requiring an independent proof of the unrestricted $B$-theorem as an additional input. For the historical relationship between the two theorems, see \cite[\S1]{GilmanSolomon}, \cite[Chapter 3]{Gorenstein}, and \cite[\S\S1.7--1.8]{ALSS}. The restriction to involutions in $G$ distinguishes our statement from some other formulations of UGT. \begin{remark} More general versions of UGT treat involutions acting from outside a simple normal subgroup. \cref{thm:UGT} is the simple-group form required for the classification induction. \end{remark} For alternating groups, three fixed points of an involution account for the nontrivial odd core. \begin{example} As noted above, the groups $A_{4a+3}$, for $a\ge1$, are unbalanced. Indeed, for $A_7$, let \[ t=(12)(34), \qquad U=\langle(567)\rangle. \] The involution $t$ fixes precisely the three points $5,6,7$. Every element of $C_{A_7}(t)$ permutes these fixed points, so conjugation preserves $U$, the alternating group on those three points. Therefore, $U$ is normal in $C_{A_7}(t)$ with order three, and consequently \[ 1\ne U\le O(C_{A_7}(t)). \] Hence $A_7$ is unbalanced, even though simplicity prevents $U$ from being normal in $A_7$ itself. The same argument works in $A_{4a+3}$. Take an involution $t$ consisting of $2a$ disjoint transpositions and fixing the remaining three points. The even number of transpositions ensures that $t\in A_{4a+3}$. The subgroup of order three generated by a $3$-cycle on the fixed points is normal in $C_{A_{4a+3}}(t)$, so this centralizer has nontrivial odd core. \end{example} \subsection{An obstruction to an accepted route to UGT} \label{sec:original} Our work on formalizing CFSG requires making the hypotheses and dependencies of its many theorems explicit. In this effort, we have assembled a comprehensive errata list. Most of the errors are straightforward to correct: typographical errors, omitted hypotheses, and similar local issues. The difficulty considered here requires a change to an argument. The first-generation proof of UGT is not contained in a single paper. Gilman--Solomon's Theorem~A \cite[p.~59]{GilmanSolomon} proves the Unbalanced Group Conjecture for a group whose proper sections satisfy it, subject to Hypothesis~S \cite[p.~58]{GilmanSolomon}, a list of standard-component problems, each asking that a group with a standard component of a specified type be identified. These problems were settled in separate papers over the following years, with the $\mathrm{PSL}_3(4)$ case completed by Cheng and Held in 1985 \cite[pp.~147--148]{ChengHeld1985}. No single reference, as far as we know, records the point at which Hypothesis~S was fully settled. Gorenstein's 1983 volume \cite{Gorenstein} develops the unbalanced-group argument; the historical overview in \cite[pp.~40--41]{GLS1} describes the collective work on the related $B_2$-property. The pattern, a result announced modulo components completed later, is discussed in \cite{AschbacherStatus}. Walter's 1986 memoir \cite{Walter} gives a treatment organized around the signalizer functor method, and is recognized by Aschbacher \cite[\S4, p.~10]{Aschbacher2013} as a route to UGT and the $B$-conjecture. It is the treatment examined here. Unexpectedly, our formalization found explicit counterexamples to a closure assertion in Walter's proof of UGT \cite{Walter}, leaving a key step of that argument unjustified. We explain the obstruction and develop an alternative argument for the simple-group case using Solomon's 1976 theory of maximal $2$-components. \begin{remark}[Other routes to UGT] \label{rem:other-routes} This difficulty does not obstruct CFSG. In particular, we have Gorenstein's account following an argument of Morton Harris \cite[\S4.5, p.~291]{Gorenstein}. There is also a proof of UGT through the second-generation work of Gorenstein, Lyons, and Solomon (GLS). Under the inductive hypothesis of \cref{thm:UGT}, the nontrivial odd-order normal subgroup places the simple group in GLS's odd-type case. Their classification of that case \cite[Part III, Chapter 12, Theorem 1.17]{GLS8} identifies it as a known group. Namely, the known-group analysis \cite[Theorem 7.7.1]{GLS3}, with these corrections \cite{GLSErrata}, then gives the required UGT family restriction. This is a short deduction from substantial second-generation input, not a repair of the obstruction described below. A shorter deduction from the reductions used in this paper is recorded in \cref{rem:shorter-formalized-route}. \end{remark} We outline the proof route where the difficulty arises. The starting point is a nonabelian simple $\K$-proper group $G$ with an involution $t$ such that $O(C_G(t))\ne1$. The goal is to show that $G$ is in one of the families in \cref{thm:UGT}. The reductions either identify $G$ with one of the required groups or lead to a configuration involving two commuting involutions and a $2$-component of one of their centralizers. A $2$-component is a perfect subnormal subgroup whose quotient by its odd core is quasisimple; after further quotienting by the center, one obtains a nonabelian simple group (see \cref{sec:preliminaries}). The task is to find a $2$-component whose structure and position in $G$ allow a recognition theorem to identify $G$. To do this, the proof compares $2$-components in different involution centralizers and selects one that is maximal with respect to these comparisons. Results of Thompson--Burgoyne and Aschbacher, together with the alternating-group analyses, handle many of the possibilities. The remaining cases require the work of Foote, Gilman--Solomon, Griess--Solomon, and Harris \cite{Burgoyne,Aschbacher1977,Foote,GilmanSolomon, GriessSolomon,Harris,ALSS}. The difficulty arises in the comparison stage. The argument works with components whose associated simple quotients belong to a specified family. A later step requires a quotient obtained by taking an involution centralizer to remain in that family. The closure assertion used to justify this can fail. Namely, an involutory automorphism can have a centralizer with a component whose simple quotient lies outside the chosen family. Therefore, the inference placing the resulting component back in the family is unjustified. This assertion is in John Walter's 1986 memoir \cite{Walter}, in the preparation for Part~II, Proposition~5.9(ii). The proposition appears on p.~145, with its proof on pp.~145--148. The passage on p.~136 following (5.2) introduces the family $\mathfrak L_0$ consisting of the groups $\PSL_2(q)$ for odd prime powers $q$ other than $9$, Fermat primes, and Mersenne primes. A Fermat prime has the form $2^{2^n}+1$ for an integer $n\geq0$, and a Mersenne prime has the form $2^n-1$ for an integer $n\geq2$. The obstruction comes from field involutions, automorphisms induced by the order-two automorphism of a quadratic extension of finite fields. For $\PSL_2(r^2)$, this automorphism raises each matrix entry to its $r$th power. The following proposition shows how a group in $\mathfrak L_0$ can have a field-involution centralizer whose component lies outside $\mathfrak L_0$. \begin{proposition}[Field-involution centralizers] \label{prop:field} Let $r\ge5$ be an odd prime power and let $\sigma$ be the automorphism of $S=\PSL_2(r^2)$ induced by $x\mapsto x^r$ on matrix entries. Then we have \[ C_S(\sigma)\cong\PGL_2(r),\qquad E(C_S(\sigma))=\Ltwo(C_S(\sigma))\cong\PSL_2(r). \] Here $E$ and $\Ltwo$ denote the ordinary and $2$-layers, defined in \cref{sec:preliminaries}. \end{proposition} \begin{proof} Put $k=\mathbf F_r$ and $F=\mathbf F_{r^2}$, the fields with $r$ and $r^2$ elements. A projective matrix $[A]\in\PGL_2(F)$ is fixed by $\sigma$ precisely when $A^\sigma=\lambda A$ for a scalar $\lambda$. Divide $A$ by any nonzero entry $a$. Since $a^r=\lambda a$, the matrix $A/a$ is fixed entrywise, and has entries in $k$. Hence $C_{\PGL_2(F)}(\sigma)=\PGL_2(k)$. Every element of $k^\times$ is a square in $F^\times$. For example, if $g$ generates $F^\times$, then $k^\times=\langle g^{r+1}\rangle$, and $r+1$ is even. Therefore, every projective matrix over $k$ has square determinant over $F$, and the entire fixed group lies in $\PSL_2(F)$. This proves the centralizer assertion. The subgroup $\PSL_2(r)$ is nonabelian simple, perfect, and normal of index two in $\PGL_2(r)$. Any odd normal subgroup of the latter maps trivially to this quotient of order two, hence lies in $\PSL_2(r)$ and is trivial. The derived subgroup is $\PSL_2(r)$, so this is its unique component and its unique $2$-component. \end{proof} \cref{prop:field} shows why the proposed closure fails. Start with $S=\PSL_2(49)$, which belongs to $\mathfrak L_0$. The number $49$ is an odd prime power, is not $9$, and is not a prime. The field involution fixes a subgroup isomorphic to $\PGL_2(7)$. The simple component of that subgroup is $\PSL_2(7)$, which does \emph{not} belong to $\mathfrak L_0$, because $7$ is a Mersenne prime. Therefore, taking the component of an involution centralizer can carry us outside the chosen family. The same calculation gives the following three examples; the middle column records the simple component of the field-involution centralizer. \medskip \begin{center} \begin{tabular}{|lll|} \toprule Group in $\mathfrak L_0$ & Centralizer component & Reason it is excluded\\ \midrule $\PSL_2(49)$ & $\PSL_2(7)$ & $7$ is a Mersenne prime\\ $\PSL_2(81)$ & $\PSL_2(9)$ & $9$ is excluded explicitly\\ $\PSL_2(289)$ & $\PSL_2(17)$ & $17$ is a Fermat prime\\ \bottomrule \end{tabular} \end{center} \medskip \noindent Here the involution is an automorphism of $S$. It can be realized as an element of the larger group $S\rtimes\langle\sigma\rangle$; it is not an involution belonging to $S$ itself. We explain where the closure assertion enters Walter's proof. The argument compares components in different involution centralizers and needs to show that a newly encountered component still belongs to a specified family. The \dfn{type} of a component is the isomorphism type of its nonabelian simple quotient, after removing its odd core and center. Knowing that two components have the same type transfers membership in a family only if one already knows that this common type belongs to that family. On p.~146 of \cite{Walter}, the relevant family is denoted by $\mathscr M_2$ and consists of components of $\mathfrak L_0$-type. Starting from a component $L_t$, Walter forms \[ L_{t,s}=L(C_{L_t}(s)), \] the layer of the centralizer of the involution $s$ in $L_t$, using his notation for the layer. In the branch $M_s^t\ne M_s$, that is, when conjugation by $t$ sends $M_s$ to a different component, the proof identifies the type of $M_s$ with that of $L_{t,s}$ and uses this identification to place $M_s$ in $\mathscr M_2$. The missing justification is the assertion that the type of $L_{t,s}$ still belongs to $\mathfrak L_0$. Membership of the starting type in $\mathfrak L_0$ does not establish this. Indeed, the calculation above shows that taking an involution centralizer can produce an excluded type. Equality of the types of $M_s$ and $L_{t,s}$ cannot supply the missing membership. For instance, if the resulting type is $\PSL_2(7)$, identifying another component with that type does not put either one back in $\mathfrak L_0$. The subsequent argument within $\mathscr M_2$ therefore requires an additional justification or a different comparison argument. The scope of these counterexamples deserves emphasis. \begin{remark} The examples above refute the closure assertion used in the proof of Proposition~II.5.9(ii). They do not provide groups that violate its conclusion. Namely, we have identified a gap in this inference, not a counterexample to the proposition itself or to UGT. \end{remark} Here we give an alternative argument. Rather than assuming that the list $\mathfrak L_0$ survives every comparison, it keeps track of the actual components and the involutions used to compare them. Under suitable maximality and rank hypotheses, we prove that the relevant comparisons preserve the simple quotient. The precise statement is \cref{thm:P1} below, which is the main theorem of this paper. Solomon's theory of maximal $2$-components \cite{Solomon} supplies the method for this comparison argument. The maximality reductions come from Gilman--Solomon \cite{GilmanSolomon}, and the final recognition uses the work of Foote and Harris as described in \cref{sec:proof}. We develop these arguments with the witnessing involutions and the passages through odd-core quotients made explicit. Our description of the approach as first-generation refers to this component-comparison and recognition strategy. For a systematic exposition of component calculus and known-group calculations, we also cite the later volumes of Gorenstein, Lyons, and Solomon \cite{GLS2,GLS3}. The date of an exposition should be distinguished from the provenance of the results it presents. Our GLS citations identify the precise statements used below; they do not constitute an appeal to the separate second-generation odd-type classification route described in \cref{rem:other-routes}. \cref{thm:P1} is proved in Sections~3--8, and we apply it to UGT in \cref{sec:proof}. Under its maximality and rank hypotheses, that theorem shows that the allowed comparisons preserve the simple top. In \cref{sec:proof}, we combine this control with the reductions of Gilman--Solomon and the recognition theorems of Foote and Harris to deduce the simple-group case of UGT. This is the route formalized by AxiomProver. The same reductions also yield a shorter deduction of UGT, since the GS-maximality they establish already satisfies Foote's hypothesis; see \cref{rem:shorter-formalized-route}. On the other hand, \cref{thm:P1} is also stronger than what is needed for UGT and we believe it to be of independent interest. Thus we choose to structure this paper around the proof of the full \cref{thm:P1}. \section{Preliminaries} \label{sec:preliminaries} The comparison argument uses components, their simple quotients, and the involutions through which they are compared. We introduce these notions, state the comparison theorem, and collect its basic inputs. Throughout, all groups are finite. A group of order a power of two is a \dfn{$2$-group}. We write $S_n$ and $A_n$ for the symmetric and alternating groups on $\{1,\ldots,n\}$, respectively. We use $O(X)$ for the largest normal subgroup of $X$ of odd order, and $O_2(X)$ for its largest normal $2$-subgroup, called its \dfn{$2$-core}. The trivial group and its identity element are both denoted by $1$. For subgroups $A,B\leq X$, we have the commutator subgroup \[[A,B]=\langle a^{-1}b^{-1}ab:a\in A,\ b\in B\rangle.\] For an element $b$, write $[A,b]=\langle a^{-1}b^{-1}ab:a\in A\rangle$. Write $X'=[X,X]$, and define the derived series by $X^{(0)}=X$ and $X^{(i+1)}=(X^{(i)})'$. Its eventual constant term is the \dfn{perfect residual} $X^{(\infty)}$. A group $X$ is \dfn{perfect} if $X'=X$ and \dfn{solvable} if $X^{(\infty)}=1$. Every perfect subgroup of $X$ is contained in $X^{(\infty)}$. A subgroup $J$ is \dfn{subnormal} in $X$ if there is a chain $J=J_0\unlhd J_1\unlhd\cdots\unlhd J_n=X$. We write $Z(X)$ for the center of $X$. A group $J$ is \dfn{quasisimple} if it is perfect and $J/Z(J)$ is nonabelian simple. A \dfn{covering group} of a nonabelian simple group $T$ is a quasisimple group $Q$ with $Q/Z(Q)\cong T$. It is a \dfn{perfect double cover} if $|Z(Q)|=2$. An \dfn{ordinary component} of $X$ is a subnormal quasisimple subgroup. Distinct ordinary components commute; their product is the \dfn{layer} $E(X)$. A group that is a product of commuting quasisimple subgroups is \dfn{semisimple}. Such a product is a central product: each factor meets the product of the others in its center. A \dfn{$2$-component} of $X$ is a perfect subnormal subgroup $J$ for which $J/O(J)$ is quasisimple. A \dfn{local $2$-component} of $G$ is a $2$-component of $C_G(a)$ for some involution $a$. When distinguishing these from components of $G$ itself, we call the latter \dfn{global components}. The product of the $2$-components of $X$ is the \dfn{$2$-layer} $\Ltwo(X)$. Each $2$-component is normal in the $2$-layer. Distinct $2$-components commute modulo $O(X)$, although they need not commute in $X$. Define \[Z^*(X)/O(X)=Z(X/O(X))\quad\text{and}\quad\tau(X)=X/Z^*(X).\] Thus, $Z^*(X)$ is the inverse image of $Z(X/O(X))$. If $J$ is a $2$-component, then $\tau(J)$ is nonabelian simple. We call $\tau(J)$ its \dfn{simple top}, and the isomorphism class of $\tau(J)$ its \dfn{type}. For a group $Q$, a group $M$ is \dfn{odd-by-$Q$} if $M/O(M)\cong Q$. The group $O^{2'}(X)$ is the smallest normal subgroup of $X$ with odd-order quotient. Equivalently, it is generated by the elements of $X$ whose orders are powers of two. Such elements are called \dfn{$2$-elements}. The symbols $O^{2'}(X)$ and $O_{2'}(X)=O(X)$ have different meanings. An \dfn{elementary abelian $2$-group} has the form $C_2^d$; the \dfn{$2$-rank} $m_2(X)$ is the largest rank of such a subgroup of $X$. A \dfn{fourgroup} is a group isomorphic to $C_2^2$, and a \dfn{line} in an elementary abelian $2$-group is a subgroup of order two. We write $\Inv(X)$ for the set of involutions of $X$ and $U^\#=U\setminus\{1\}$. For an abelian $2$-group $P$, let $\Omega_1(P)=\{x\in P:x^2=1\}$. We write $\Syl_2(X)$ for the set of Sylow $2$-subgroups of $X$ and $|X|_2$ for the largest power of two dividing $|X|$. For subsets $A,B\subseteq X$, write $C_X(A,B)=C_X(A)\cap C_X(B)$, allowing either argument to be a single element. Conjugation is written $x^g=g^{-1}xg$ and $M^g=g^{-1}Mg$. We write $N_X(M)$ for the normalizer of $M$ in $X$. A \dfn{$2$-local subgroup} of $X$ is $N_X(Q)$ for a nontrivial $2$-subgroup $Q\leq X$. For subgroups $A,B$, the notation $\langle A^B\rangle$ denotes the group generated by $A^b$ for $b\in B$. In particular, $\langle A^X\rangle$ is the normal closure of $A$ in $X$ when $A\leq X$. For $M\leq X$, put \[\CC_X(M)=\{g\in N_X(M):[M,g]\leq O(M)\}.\] This is the kernel of the action of $N_X(M)$ on $M/O(M)$. For a perfect group $M$ such that $M/O(M)$ is quasisimple, and a $2$-subgroup $U\leq\CC_G(M)$, write \[M_U=C_M(U)^{(\infty)}\quad\text{and}\quad M_u=M_{\langle u\rangle}.\] We will repeatedly use that $M_UO(M)=M$; this is proved in \cref{lp:lem:fixed-layer}. We say that a subgroup $D\leq M$ \dfn{covers} $M/O(M)$ if $DO(M)=M$. The symbols $\Aut(X)$ and $\operatorname{Inn}(X)$ denote the automorphism group and its subgroup of inner automorphisms, and $\Out(X)=\Aut(X)/\operatorname{Inn}(X)$. A finite group is \dfn{nilpotent} if it is the direct product of its Sylow subgroups. The \dfn{Fitting subgroup} $F(X)$ is the largest nilpotent normal subgroup of $X$. The \dfn{generalized Fitting subgroup} is $F^*(X)=F(X)E(X)$. We use the standard theorem \[C_X(F^*(X))\leq F^*(X)\] from \cite[Theorem~3.6]{GLS2}. In particular, if $O(X)=1$, then $F(X)=O_2(X)$. Finally, $O_{2',2}(X)$ is given by \[O_{2',2}(X)/O(X)=O_2(X/O(X)).\] \subsection{Comparison relations} To compare the $2$-components arising in different involution centralizers, we adapt the correspondence of Gilman--Solomon (GS) \cite[p.~56]{GilmanSolomon} and the maximality framework of Foote \cite{Foote}. Our GS relation uses a containment in place of Gilman--Solomon's layer equality; \cref{lp:lem:gs-criterion} explains the passage between these forms where it is needed. The formulations below record the witnessing involutions explicitly and distinguish the pointed and mirror steps used in our argument. \begin{definition} \label{def:relations} An \dfn{occurrence} is a pair $(a,J)$ in which $a$ is an involution and $J$ is a $2$-component of $C_G(a)$. Suppose that $(a,J)$ and $(b,K)$ are occurrences, $a$ and $b$ commute, and $b\in\CC_G(J)$. A \dfn{GS correspondence}, with these witnesses, is the inclusion \begin{equation} J_b\le\langle K,K^a\rangle. \label{eq:corr} \end{equation} It is \dfn{isomorphic} if $K^a=K$ and $J_bO(K)=K$. The fixed-layer lemma identifies $J_b$ with $\Ltwo(C_J(b))$. A \dfn{pointed Foote step} $(a,J)\longrightarrow_F(b,K)$ requires the same hypotheses, $a\ne b$, and the equality \begin{equation} \left\langle \left(O^{2'}(C_J(b)O(C_G(b)))\right)^{\Ltwo(C_G(b))} \right\rangle=\langle K,K^a\rangle. \label{eq:pointed} \end{equation} An occurrence is \dfn{pointed-maximal} if every finite chain of pointed Foote steps starting there preserves the isomorphism type of its simple top. The endpoint occurrence of each step must be the starting occurrence of the next. A local $2$-component is \dfn{GS-maximal} if every forward chain of GS correspondences preserves its simple top, with the witnessing involutions allowed to be chosen independently at successive steps. A \dfn{mirror step} $J\longrightarrow_M K$ has commuting involutions $z,s$, with $J$ a $2$-component of $C_G(z)$, $J^s\ne J$, $K$ a $2$-component of $C_G(s)$, and a $2$-component $D$ of $C_G(z)\cap C_G(s)$ satisfying \[ D\le\langle J,J^s\rangle,\qquad D\le\langle K,K^z\rangle. \] Write $J\rightsquigarrow K$ for a finite chain of GS correspondences, mirror steps, and conjugations, allowing the empty chain. A local $2$-component is \dfn{mirror-maximal} if every continuation preserves its simple top. \end{definition} A subgroup of a direct product of groups is \dfn{diagonal} on a set of factors if it projects isomorphically onto each of those factors and trivially onto the others. For example, the automorphism exchanging the factors of $T\times T$ fixes the diagonal $\{(x,x):x\in T\}$, which is isomorphic to either factor but is neither. The \dfn{support} of a subgroup of a direct product is the set of factors on which it has nontrivial projection. The small simple types remaining in the UGT reduction lead to the following hypotheses. Their rank condition will be verified in \cref{sec:proof}. \begin{hypothesis} \label{hyp:ambient} The group $G$ is nonabelian simple and $\K$-proper. Fix a simple group $T$ isomorphic to $A_7$ or to $\PSL_2(q)$ for an odd prime power $q\geq5$, and put \[ \mathcal F_T=\{A\leq G:\tau(A)\cong T\text{ and }(z,A)\text{ is pointed-maximal for some }z\in\Inv(G)\}. \] Assume that every member of $\mathcal F_T$ has $2$-rank two. \end{hypothesis} For the comparison argument in the remainder of Sections~2--8, we assume \cref{hyp:ambient}, with the following exceptions. The fixed-layer lemma \cref{lp:lem:fixed-layer} holds under its own stated hypotheses, and the known-group assertion in \cref{lp:lem:local-b} requires only that $H$ be a $\K$-group. The correspondence lemma \cref{lp:lem:gs-foote}, the rank-increase lemma \cref{lp:lem:rank-increase}, and the maximality criterion \cref{lp:lem:gs-criterion} do not require the assumption on $\mathcal F_T$; we retain the assumption that the ambient $G$ is nonabelian simple and $\K$-proper, together with the hypotheses stated in each lemma. In particular, the rank-increase lemma requires $m_2(J)\ge2$ for its source component, and the maximality criterion requires the stated local $B$-property. These results are therefore available in \cref{sec:proof} before the family-rank hypothesis is established. Under the full hypotheses of \cref{hyp:ambient}, pointed maximality controls the larger collection of comparisons, including mirror steps. \begin{theorem}[Preservation of simple tops] \label{thm:P1} \label{thm:main} Assume \cref{hyp:ambient}. If $A\in\mathcal F_T$ and $A\rightsquigarrow J$, then $\tau(J)\cong T$. In particular, every member of $\mathcal F_T$ is mirror-maximal. \end{theorem} We first show that every member of $\mathcal F_T$ can be reached from a GS-maximal component of type $T$. We then exclude a GS-maximal component of type $T$ that is interchanged with another component by a commuting involution. The latter argument uses a family of subgroups closed under isomorphic correspondences, a signalizer construction, and an embedding theorem. \subsection{Component calculus} \label{lp:sec:background} The following facts govern passage to odd-core quotients and to centralizers of $2$-subgroups. We use the component calculus as presented by Gorenstein, Lyons, and Solomon \cite[\S5]{GLS2}, together with the coprime-action results in Gorenstein's \emph{Finite Groups} \cite[Chapter~5]{GorensteinFinite}. If $R=O(X)$, the quotient map gives a bijection between the $2$-components of $X$ and the ordinary components of $X/R$. The inverse image of a component has a unique $2$-component, namely its perfect residual. We call this subgroup its \dfn{canonical perfect lift}. A subnormal subgroup remains subnormal after intersection with another subgroup, and the image of a $2$-component under a surjective homomorphism with odd kernel is a $2$-component. We also use \dfn{$L_{2'}$-balance}: if a $2$-group $U$ acts on $X$, then $\Ltwo(C_X(U))\leq\Ltwo(X)$. In its comparison form, if $a,b$ are commuting involutions and $J$ is a $2$-component of $C_G(a)$ with $b\in\CC_G(J)$, then there is a $2$-component $K$ of $C_G(b)$ such that $J_b\leq\langle K,K^a\rangle$. See \cite[Theorem~5.24]{GLS2} and \cite[p.~56]{GilmanSolomon}. Thus, the comparison relation in \cref{def:relations} always has a target for these witnesses. An action of $U$ on $R$ is \dfn{coprime} if $|U|$ and $|R|$ are relatively prime. For such an action, we use $R=[R,U]C_R(U)$ and $[R,U,U]=[R,U]$. If $U$ is a fourgroup and $R$ has odd order, then \[ R=\langle C_R(u):u\in U^\#\rangle. \] We use fixed-point lifting. Namely if a $2$-group $U$ acts on $X$ and preserves an odd-order normal subgroup $R$, then \[C_X(U)R/R=C_{X/R}(U).\] This follows from conjugacy of complements to $R$ in $R\rtimes U$ by the Schur--Zassenhaus theorem \cite{GorensteinFinite}. By the odd-order theorem \cite{FeitThompson}, odd-order groups are solvable. Consequently, an odd-order perfect group is trivial. These facts also give a useful uniqueness statement: if $P\unlhd X$ is perfect and $X/P$ is solvable, then $X^{(\infty)}=P$. If an automorphism $\alpha$ of a perfect group $P$ acts trivially on $P/Z(P)$, then it is the identity. Indeed, the map $x\mapsto x^{-1}x^\alpha$ is a homomorphism from $P$ to the abelian group $Z(P)$ and is therefore trivial. Also, if a $2$-group $U$ acts on a group $X$ of even order, then $C_X(U)$ has even order: every nontrivial $U$-orbit has even size. We use the three-subgroups lemma in the following form. Write $[A,B,C]=[[A,B],C]$. If $R\unlhd\langle A,B,C\rangle$ and two of $[A,B,C]$, $[B,C,A]$, and $[C,A,B]$ are contained in $R$, then so is the third. Fixed-point lifting supplies the perfect subgroups used in our comparisons: they retain the full quotient by the odd core. \begin{lemma}[Fixed layers]\label{lp:lem:fixed-layer} Let $M$ be perfect with $M/O(M)$ quasisimple, and let a $2$-group $U$ normalize $M$ and act trivially on $M/O(M)$. Then we have \[ C_M(U)O(M)=M,\qquad M_U/O(M_U)\cong M/O(M),\qquad C_M(U)/M_U\text{ has odd order}. \] In particular $M_U=O^{2'}(C_M(U))$ and it is the unique $2$-component of $C_M(U)$. If $M$ is a $2$-component of $C_G(a)$ and $u$ is an involution commuting with $a$ and in $\CC_G(M)$, then $M_u$ is a $2$-component of $C_G(a)\cap C_G(u)$. \end{lemma} \begin{proof} Put $R=O(M)$ and $C=C_M(U)$. Fixed-point lifting gives $CR=M$. Since $M/R$ is perfect, $M_UR=M$, and hence \[ M_U/(M_U\cap R)\cong M/R,\qquad C=M_U(C\cap R). \] The group $M/R$ has trivial odd core, so $O(M_U)=M_U\cap R$. The quotient $C/M_U$ has odd order; conversely, the perfect group $M_U$ has trivial image in $C/O^{2'}(C)$ by the odd-order theorem. Thus, $M_U=O^{2'}(C)$, and the component correspondence gives uniqueness. Finally, intersecting a subnormal chain for $M$ in $C_G(a)$ with $C_G(u)$ makes $C_M(u)$, and hence $M_u$, subnormal in $C_G(a)\cap C_G(u)$. \end{proof} We first record the covering-group facts independently of the family-rank hypothesis. If $Q$ is perfect, $O(Q)=1$, and $Q/Z(Q)\cong T$, where $T\cong A_7$ or $\PSL_2(q)$ for odd $q\ge5$, then $Q$ is either $T$ or its perfect double cover. In the latter case its Sylow $2$-subgroups have a unique involution, so $m_2(Q)=1$; in the former case $m_2(Q)=2$. The odd central factors in the covers of $A_6$ and $A_7$ are excluded by $O(Q)=1$. These covering-group and Sylow-subgroup calculations are recorded in \cite[Section~6.1]{GLS3}. The next lemma uses the family-rank hypothesis to exclude the double-cover possibility. \begin{lemma}\label{lp:lem:small-cover} If $A\in\mathcal F_T$, then $A/O(A)\cong T$. The same conclusion holds for every GS-maximal local $2$-component of type $T$. \end{lemma} \begin{proof} Quotienting by an odd-order normal subgroup preserves $2$-rank: the inverse image of an elementary abelian $2$-subgroup has a Sylow $2$-subgroup mapping isomorphically onto it. Apply the preceding covering-group facts to $Q=A/O(A)$. The rank-two hypothesis in \cref{hyp:ambient} excludes the double cover, so $A/O(A)\cong T$. Every pointed step is a GS correspondence, directly from its defining equality and \cref{lp:lem:fixed-layer}. Thus, a GS-maximal component of top $T$ is pointed-maximal at every occurrence and belongs to $\mathcal F_T$. \end{proof} The involutions and fourgroups of $T$ supply the configurations used in the comparison and signalizer arguments. \begin{lemma}\label{lem:small-top-facts} The group $T$ has one conjugacy class of involutions, every involution belongs to a fourgroup, and every involution centralizer is solvable. Its outer automorphism group is solvable. If $V\leq T$ is a fourgroup, then $N_T(V)$ acts transitively on $V^\#$. \end{lemma} \begin{proof} A dihedral group of order $2n$ is the group with generators $r,s$ and relations $r^n=s^2=1$ and $srs=r^{-1}$; these groups are solvable. For $A_7$, the involutions are the double transpositions. They form one conjugacy class, and their centralizers are subgroups of the product of a group of order eight and $S_3$, hence are solvable. For $\PSL_2(q)$ with $q$ odd, the matrix description gives one involution class, dihedral Sylow $2$-subgroups, and dihedral involution centralizers. These Sylow subgroups have order at least four. The Sylow $2$-subgroups of $A_7$ are also dihedral of order eight, so every involution of either group belongs to a fourgroup. Finally, $\Out(A_7)$ has order two. Writing $q=p^f$ with $p$ prime, $\Out(\PSL_2(p^f))$ is a product of a cyclic group of order two and a cyclic group of order $f$. It remains to verify the assertion about $V$. A fourgroup in $A_7$ has either one orbit of size four and three fixed points, or has three orbits of size two and one fixed point. In the first case the alternating group on the four points supplies an element inducing a $3$-cycle on $V^\#$. In the second case an even permutation cyclically permuting the three two-point orbits does so. Now let $T=\PSL_2(q)$, and write $I$ for the identity matrix. The inverse image of $V$ in the group $\operatorname{SL}_2(q)$ of determinant-one matrices has generators $i,j$ satisfying $i^2=j^2=-I$ and $ij=-ji$. Put $k=ij$ and $w=(I+i+j+k)/2$. The displayed relations give $\det(w)=1$ and show that conjugation by $w$ cyclically permutes the projective images of $i,j,k$. Thus, the image of $w$ normalizes $V$ and acts transitively on $V^\#$. \end{proof} The inductive hypothesis supplies the following local $B$-property inside proper subgroups. Its use does not require the $B$-property for $G$. \begin{lemma}[$B$-property for known groups]\label{lp:lem:local-b} Every $\K$-group $H$ satisfies \begin{equation}\label{eq:localB} [\Ltwo(N_H(Q)),O(N_H(Q))]\leq O(H) \end{equation} for every nontrivial $2$-subgroup $Q\leq H$. If $O(H)=1$, the $2$-components of its $2$-local subgroups are ordinary. Consequently, every proper subgroup of the ambient $G$ satisfies \eqref{eq:localB}, and the proper-subgroup $B$ hypothesis in Gilman--Solomon's Lemma~3.5 is available. \end{lemma} \begin{proof} Suppose first that $O(H)=1$. The known-group fixed-point result \cite[Lemma~5.28]{GLS2}, with its hypotheses supplied by \cite[Theorem~7.1.9]{GLS3}, says that every $2$-component of $C_H(Q)$ is quasisimple. Put $M=N_H(Q)$ and let $J$ be a $2$-component of $M$. Modulo $O(M)$ its component image centralizes the normal $2$-subgroup $QO(M)/O(M)$. Hence \[ [J,Q]\le O(M)\cap Q=1. \] Intersecting a subnormal chain with $C_H(Q)$ makes $J$ a $2$-component of $C_H(Q)$, so the cited result makes $J$ quasisimple. It is therefore an ordinary component of $M$. Ordinary components centralize the odd core of $M$, so the commutator in \eqref{eq:localB} is trivial in this case. For arbitrary $H$, quotient by $R=O(H)$. The map \[ N_H(Q)\longrightarrow N_{H/R}(QR/R) \] is onto with odd kernel. Indeed, an element normalizing $QR/R$ can be adjusted by an element of $R$ to normalize the Sylow $2$-subgroup $Q$ of $QR$. The image of a $2$-component under this odd-kernel map is a $2$-component, and the image of the odd core is contained in the odd core of the image. The case with trivial odd core gives \eqref{eq:localB}. Finally every nontrivial $2$-local subgroup of a nonabelian simple group is proper, so $\K$-properness supplies the hypotheses. The right-hand side in \eqref{eq:localB} is $O(H)$; we have not asserted the global B-property for the unknown group $G$. \end{proof} \section{Comparing Components} \label{lp:sec:transport} The maximality argument in this section adapts Gilman--Solomon's rank-increase lemma and maximality criterion \cite[Lemmas~3.4--3.5]{GilmanSolomon}. Our goal is to reach each pointed-maximal component from a GS-maximal one. We first relate the GS and pointed Foote correspondences with witnesses fixed, then use mirror steps to reverse the comparisons needed for this choice. \begin{lemma}\label{lp:lem:gs-foote} A GS correspondence with distinct witnesses $a,b$ is a pointed Foote step with the same witnesses. Conversely, every pointed Foote step is a GS correspondence. A GS correspondence with $a=b$ has $J=K$. \end{lemma} \begin{proof} Put $D=J_b$, $E=\Ltwo(C_G(b))$, $R=O(C_G(b))$, and $W=\langle K,K^a\rangle$. The GS condition is $D\leq W$. The group $D$ is a $2$-component of $C_G(a)\cap C_G(b)$, and $C_J(b)/D$ has odd order. Modulo $R$, its image has noncentral projection on a component represented by $W$. If $K^a\ne K$, then $a$ fixes $D$ pointwise and interchanges the two factors, so both projections are noncentral. Taking normal closures in the layer, and then the unique $2$-component lifts, gives $\langle D^E\rangle=W.$ In particular $C_J(b)$ normalizes $W$. The group $R$ also normalizes every $2$-component of $C_G(b)$, hence normalizes $W$. The quotient $C_J(b)RW/W$ has odd order, since both the image of $C_J(b)/D$ and the normal image of $R$ have odd order. Consequently \[ F=O^{2'}(C_J(b)R)\leq W. \] On the other hand $D\leq F$, because $D=O^{2'}(D)$. Thus, $\langle F^E\rangle=W$, the required pointed equality. The converse follows from $D\leq F$. If $a=b$, then $D=J$ and $K^a=K$, so $J\leq K$; two $2$-components of the same group cannot be properly nested. \end{proof} Equality of the simple quotients permits a return to the source, using a mirror step when two target components are interchanged. \begin{lemma}\label{lp:lem:reverse} Suppose $(a,J)\longrightarrow_F(b,K)$ and $J/O(J)\cong K/O(K)\cong T$. Then $K\rightsquigarrow J$. If the correspondence is isomorphic, its reverse is also isomorphic. \end{lemma} \begin{proof} Let $D=J_b$. If $K^a\ne K$, the common-centralizer component $D$ supplies the reverse mirror step, with witnesses $(b,a)$: \[ D\leq\langle K,K^a\rangle,\qquad D\leq J=\langle J,J^b\rangle. \] If $K^a=K$, then $D\leq K$ and $D\cap O(K)\leq O(D)$. Since $D/O(D)\cong T\cong K/O(K)$, comparison of orders gives $DO(K)=K$ and $D\cap O(K)=O(D)$. Thus, $a$ acts trivially on $K/O(K)$. Intersecting a subnormal chain for $D$ in the common centralizer with $C_K(a)$ shows that $D$ is its unique $2$-component. Therefore $K_a=D\leq J$, which is the reverse GS correspondence. Moreover $DO(J)=J$, so this reverse correspondence is isomorphic. \end{proof} \subsection{Increasing the centralizer rank} The maximal choices below rely on an increase in quotient order or centralizer rank. Starting from a nonisomorphic correspondence, we can obtain such an increase while keeping the source witness $a$ fixed, although the second witness may change. This adapts \cite[Lemma~3.4, pp.~68--69]{GilmanSolomon}, using only $\langle a,b\rangle\le S$. \begin{lemma}[Rank increase] \label{lp:lem:rank-increase} Let $J$ be a $2$-component of $C_G(a)$ with $m_2(J)\ge2$. Let \[ S\in\Syl_2(C_G(a)\cap N_G(J)),\qquad P=S\cap\CC_G(J),\qquad b\in\Inv(P). \] If a correspondence from $(a,J)$ at $b$ is not isomorphic, there are $e\in\Inv(P)$ and a correspondent $K$ at $(a,e)$ with $C_S(b)\le C_S(e)$ such that either \[ |K/O(K)|>|J/O(J)|, \] or \[ |K/O(K)|=|J/O(J)|,\qquad m_2(C_G(e)\cap\CC_G(K))> m_2(C_G(a)\cap\CC_G(J)). \] \end{lemma} \begin{proof} Choose $e\in\Inv(P)$ admitting a nonisomorphic correspondent and such that $Q=C_S(e)$ contains $C_S(b)$ and is maximal among these centralizers. Write $K$ for a nonisomorphic correspondent at $e$. The correspondence gives $|K/O(K)|\ge|J/O(J)|$; see \cite[Lemma~2.1(iv)]{Foote}. If the inequality is strict, we are done. Suppose equality holds, and put $T_0=J/O(J)$. Then $K^a\ne K$: otherwise $J_e\le K$ and order comparison makes the correspondence isomorphic. Set $M=C_G(e)$ and use bars for images modulo $O(M)$. The two components give \[ \overline{\langle K,K^a\rangle}=\bar K\times\bar K^a, \qquad \bar K\cong T_0. \] These assertions and the diagonal description of $\bar J_e$ follow from the equal-order case of the correspondence calculation \cite[Lemma~3.4, proof]{GilmanSolomon}. To see why the product is direct, a nontrivial common center would contain an involution fixed by $a$ and hence in the kernel of the diagonal map. This would make $J_e/O(J_e)$ a proper central quotient of $K/O(K)$, contrary to equality of orders. Put $P_0=P\cap Q=C_P(e)$. This group centralizes $J_e/O(J_e)$, and therefore its permutation action on the two factors has kernel \[ R=C_{P_0}(\bar K\times\bar K^a),\qquad P_0=\langle a\rangle\times R. \] Here $a$ is central in $S$, and the assertion about the kernel follows because an automorphism fixing both factors and the diagonal pointwise fixes both factors pointwise. The opposite factor $\bar K^a$ and $\bar R$ centralize $\bar K$ and intersect trivially: $a$ centralizes $\bar R$ and interchanges the two direct factors. Lifting $2$-subgroups through the odd kernel gives \[ m_2(C_G(e)\cap\CC_G(K)) \geq 2+m_2(R)>m_2(P_0). \] Thus, it suffices to show $m_2(P_0)=m_2(P)$. Suppose otherwise. Since $P\trianglelefteq S$, the product $PQ$ is a $2$-group, and $P_0Q,\qquad N_{PQ}(Q)=N_P(Q)Q. \] Consequently $N_P(Q)>P_0$, and $P_0\trianglelefteq N_P(Q)$. Choose $w\in N_P(Q)\setminus P_0$ whose image in $N_P(Q)/P_0$ is an involution. Then $w\notin Q$ and $w^2\in P_0$. We have $|P_0|\geq4$, since $a,e$ are distinct commuting involutions. If $|P_0|=4$, use the elementary fact from \cite[Lemma~3.4]{GilmanSolomon}: a finite $2$-group with an involution whose centralizer has order four has $2$-rank at most two. This contradicts $m_2(P_0)\rho_0$. The triple $(e,K,a)$ is an interchanged triple, contrary to the definition of $\rho_0$. Isomorphic correspondences reverse by \cref{lp:lem:reverse}. Let $\mathcal I(Y)$ be the set of local $2$-components joined to $Y$ by a finite chain of isomorphic GS correspondences, where the witnessing involutions may change between edges. Reversibility makes this an equivalence class; every member is GS-maximal of type $T$. Optimize $(\rho(a,J),\nu(a,J))$ over all $J\in\mathcal I(Y)$ and all involutions $a$ for which $J$ is a $2$-component of $C_G(a)$. Its first coordinate is at least $\rho_0$. The preceding paragraph gives the isomorphic correspondences needed for \cref{lp:lem:gs-criterion}; exactly the Sylow-order argument in \cref{lp:lem:optimized} supplies the remaining hypothesis. Thus, every edge in every forward GS chain from this optimized $J$ is isomorphic. There is an isomorphic path from $J$ back to $Y$. Appending any forward chain from $Y$ shows that all of its edges are isomorphic. Define \[ \mathscr S=\{K^g: g\in G,\quad Y=J_0\longrightarrow_{\rm GS}J_1\longrightarrow_{\rm GS} \cdots\longrightarrow_{\rm GS}J_n=K \text{ for some }n\geq0\}. \] The witnesses in this definition are chosen separately for each edge. Conjugation preserves the correspondence relation, so the same isomorphic-transport assertion holds from every occurrence of every member. Both $Y$ and $Y^s$ belong to $\mathscr S$, and their original interchanged triple supplies the last assertion. \end{proof} \section{Marked Subgroups} \label{lp:sec:marked} To carry the comparison information into proper subgroups, we enlarge our family to include perfect subgroups detected by its components. These will be called marked subgroups. This construction organizes the component comparisons needed for the application of Solomon's method \cite{Solomon}; the defining marking condition keeps its connection to the chosen family explicit. Fix the conjugacy-invariant family $\mathscr S$ supplied by \cref{lp:prop:IT}. Its members have quotient $T$ modulo their odd cores, and every forward GS correspondence from every occurrence of a member is isomorphic and has its target in $\mathscr S$. Recall that $M_U=C_M(U)^{(\infty)}$ for $U\leq\CC_G(M)$. By \cref{lp:lem:fixed-layer}, this subgroup covers $M/O(M)$. The construction uses the following control on normal closures: under the stated hypotheses, the image of $W$ is a single component. \begin{lemma} \label{lp:lem:normal-closure} Let $\pi:A\to A/O(A)$, and let $P\leq W\leq A$ be perfect. Suppose $\pi(P)=X$ is an ordinary component of $A/O(A)$ and \[ W=\langle P^W\rangle R \quad\text{for some odd-order normal subgroup }R\unlhd W. \] Then $\pi(W)=X$. If $X\cong T$, then $W=P\,O(W)$ and $W/O(W)\cong T$. Furthermore, a perfect odd-by-$T$ subgroup containing a simple component $X\cong T$ of its ambient group equals $X$. \end{lemma} \begin{proof} Set $V=\pi(W)$ and $N=\langle X^V\rangle$. The quotient $V/N$ is perfect and odd, hence trivial by the odd-order theorem. Each conjugate of $X$ is an ambient component, so $V=N\leq E(A/O(A))$. Components are normal in the layer; consequently $V$ normalizes $X$ and $V=X$. When $X$ is simple, $W\cap O(A)=O(W)$, which proves the assertions about $W$. For the last assertion, write $W$ for the perfect odd-by-$T$ subgroup. The simple subgroup $X$ injects into and covers $W/O(W)$, so $W=XO(W)$. Its normal closure $N=\langle X^W\rangle$ satisfies $W=NO(W)$; perfectness and the odd-order theorem give $W=N$. Again $N$ lies in the ambient layer, in which $X$ is normal. Thus, $W=X$. \end{proof} A marking records agreement with a member of $\mathscr S$ in an involution centralizer after its odd core has been removed. \begin{definition} A perfect subgroup $M\leq G$ is \dfn{marked} if $M/O(M)\cong T$ and there are an involution $a\in\CC_G(M)$ and a member $S_a\in\mathscr S$ such that $S_a$ is a $2$-component of $C_G(a)$ and \begin{equation} M_a O(C_G(a))=S_a O(C_G(a)). \label{eq:marking} \end{equation} Write $\mathscr F$ for this conjugacy-invariant family. The involution $a$ is a \dfn{marking witness}. Every member of $\mathscr S$ is marked, using any witnessing involution at which it occurs as a local component. \end{definition} The choice of a marking witness is flexible: every involution in the action kernel can serve, and nontrivial $2$-subgroups of that kernel also give a single containing component. \begin{lemma}[Transport of markings] \label{lem:ST} Let $M\in\mathscr F$. Every involution of $\CC_G(M)$ is a marking witness. For every \emph{nontrivial} $2$-subgroup $U\leq\CC_G(M)$ there is a single marked $2$-component $S_U$ of $C_G(U)$ such that \begin{equation} \left\langle M_U^{\,\Ltwo(C_G(U))}\right\rangle =S_U=O(S_U)M_U. \label{eq:ST} \end{equation} \end{lemma} \begin{proof} Suppose first that $a$ witnesses the marking, and put $R=O(C_G(a))$. Since $S_a\unlhd S_aR$ and the quotient by $S_a$ is odd, perfectness in \eqref{eq:marking} gives \[ M_a\leq S_a,\qquad S_a=O(S_a)M_a. \] Let $v\in\CC_G(M)$ be an involution commuting with $a$. It normalizes $M_aR$, hence its unique $2$-component $S_a$, and acts trivially on $S_a/O(S_a)$ because $M_a$ covers that quotient. If $v\ne a$, \cref{lp:prop:IT} supplies a single isomorphic correspondent $S_v\in\mathscr S$ at $v$. The group $M_{\langle a,v\rangle}$ lies in $(S_a)_v\leq S_v$ and still maps onto $T$. Its image in $C_G(v)/O(C_G(v))$ is therefore the simple component represented by $S_v$. The image of $M_v$ is perfect odd-by-$T$ and contains that component. \cref{lp:lem:normal-closure} makes its image exactly that component, proving \eqref{eq:marking} at $v$. The case $v=a$ needs no correspondence. For an arbitrary involution $u\in\CC_G(M)$, choose a Sylow $2$-subgroup of $\CC_G(M)$ containing $u$, and conjugate $a$ into it by an element of $\CC_G(M)$. This preserves $M$ and its marking. A central involution of that Sylow subgroup connects the conjugated witness to $u$ by two commuting steps. Hence every such $u$ is a witness. Now take $1\ne U\leq\CC_G(M)$ and $u\in\Omega_1(Z(U))^{\#}$. In $A=C_G(u)$, put $\bar A=A/O(A)$. The image of $M_u$ is a simple component $K\cong T$ of $\bar A$, and $\bar U$ centralizes $K$. Intersecting a subnormal chain shows that $K$ is a component of $C_{\bar A}(\bar U)$. Odd-kernel fixed-point lifting gives an epimorphism \[ C_G(U)=C_A(U)\longrightarrow C_{\bar A}(\bar U) \] with odd kernel. Its canonical $2$-component lift $S_U$ contains $M_U$ and satisfies $S_U=O(S_U)M_U$. Indeed, $M_U$ maps onto $K$, and every perfect subgroup of the full inverse image lies in its perfect residual. Since $u$ centralizes $S_U$ and its image in $\bar A$ is $K$, the original witness $u$ marks $S_U$ as well. Finally $S_U\unlhd\Ltwo(C_G(U))$. The normal closure of $M_U$ in that layer lies in $S_U$ and covers $S_U/O(S_U)$. Its quotient in $S_U$ is both odd and perfect, and hence trivial. This proves \eqref{eq:ST}. \end{proof} The restriction $U\ne1$ is essential to the use made of this lemma. No assertion that a marked subgroup lies in a global component of $G$ has been obtained. \subsection{Comparisons in proper subgroups} \label{lp:sec:relative} The local $B$-property gives stronger control inside proper subgroups: components with marked lifts have only identity correspondences in their odd core quotient. Fix $H