{"id":"hexagon/2610.00417","pdf":"https://hexagonmath.org/pdf/2610.00417v1","url":"https://hexagonmath.org/abs/2610.00417v1","tags":["math.GR","msc:18C15","msc:18A40","msc:20D15","adjoint tower","central extension","generalized dihedral group","involutory quandle","monadic decomposition"],"score":0,"title":"The third unit in the adjoint tower of group involutions","domain":"math.CT","origin":"ai-generated","source":"hexagon","authors":["Zhenyu Sun"],"reviews":0,"subject":"math.CT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}