Evolving groups (Q1650741)

A finite group \(G\) is called \textit{evolving} if, for every prime number \(p\) and every \(p\)-subgroup \(I\) of \(G\), there exists a subgroup \(J\) of \(G\) containing \(I\) such that \([G:J]\) is a \(p\)-power and \([J:I]\) is coprime to \(p\). Normal subgroups (but not arbitrary subgroups) and quotients of an evolving group are evolving. The author shows four main properties: A) For a finite group \(G\), the following are equivalent: (i) For every \(G\)-module \(M\), for every integer \(q\), and for every \(c\in\widehat{H}^q(G,M)\), the minimum of the set \(\{[G: H],H\leq G,\text{ with }c \in\ker\mathrm{Res}_H^G\}\) coincides with its greatest common divisor; (ii) For every \(G\)-module \(M\) and for every \(c \in \widehat{H}^0(G,M)\), the minimum of the set \(\{[G:H], H\leq G, c \in\ker\mathrm{Res}_H^G \}\) coincides with its greatest common divisor; (iii) The group \(G\) is evolving. B) A finite group \(G\) is called \textit{prime-intense} if it possesses a collection \(\{S_p\}\) of \(p\)-Sylow subgroups of \(G\) such that, for all primes \(p>q\) dividing \(|G|\), one has \(S_q\subseteq N_G(S_p)\) and, for each subgroup \(H\) in \(S_p\) and each \(x\in S_q\), the subgroups \(xHx^{-1}\) and \(H\) are \(S_p\)-conjugate. Then \(G\) is evolving if and only if it is prime-intense. C) Evolving groups are supersolvable. D) A finite group \(G\) is evolving if and only if there exist nilpotent groups \(N\) and \(T\) of coprime orders and a group homomorphism \(\varphi\colon T\to\mathrm{Int}(N)\) such that \(G=N\rtimes_\varphi T\). Note that properties A and D can be generalized to profinite groups.