math.GR — Group Theory

The Unbalanced Group Theorem

Contributed by Colin Defant, Ken Ono

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.