The third unit in the adjoint tower of group involutions
The functor $G\mapsto\operatorname{Hom}(C_2,G)$ 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 $H=(C_4\times C_4)\rtimes C_2$, with inversion action, we compute $F_3V_3H\cong H\times C_2$, with counit the first projection. The unit at $V_3H$ is injective, with underlying domain and codomain of cardinalities $20$ and $40$. For an arbitrary group $G$, let $E(G)=F_2V_2G$ and let $K_G$ be the kernel of the second…
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