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The third unit in the adjoint tower of group involutions

A mix of human-written and AI-generated textHuman understanding: all partsmath.CT — Category Theorymath.GR — Group Theory

Contributed by Zhenyu Sun ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

The functor G↦Hom⁡(C2,G)G\mapsto\operatorname{Hom}(C_2,G) determines an adjoint tower through successive Eilenberg--Moore comparison functors. We prove that the unit of the third adjunction need not be an isomorphism, answering a question of Bergman. For the generalized dihedral group H=(C4×C4)⋊C2H=(C_4\times C_4)\rtimes C_2, with inversion action, we compute F3V3H≅H×C2F_3V_3H\cong H\times C_2, with counit the first projection. The unit at V3HV_3H is injective, with underlying domain and codomain of cardinalities 2020 and 4040. For an arbitrary group GG, let E(G)=F2V2GE(G)=F_2V_2G and let KGK_G be the kernel of the second counit. We prove that KGK_G is central and that F3V3G≅E(G)/KG[2]F_3V_3G\cong E(G)/K_G[2], where KG[2]K_G[2] consists of the elements of KGK_G whose square is one. The third unit at V3GV_3G is an isomorphism precisely when KGK_G has no element of order four. A central extension of HH of order 128128 supplies such an element and yields the stated description of F3V3HF_3V_3H.

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References

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 20, 2026