Kernel systems on finite groups (Q2764669)

All groups considered are finite. For a group \(G\) and \(A\subseteq G\) we write \(A^\#=A\cap(G\setminus\{1\})\); for \((x,y)\in G\times G\) we denote \(y^x=x^{-1}yx\); and for \(A\subseteq G\) and \(x\in G\) we write \(A^x=\{y^x\mid y\in A\}\). The symbol \(\mathcal{MS}(G)\) denotes the set of maximal solvable subgroups of \(G\). A group \(G\) will be called CA (resp. CN, CS) if for each \(x\in G^\#\) the centralizer \(C_G(x)\) is Abelian (resp. nilpotent, solvable).NEWLINENEWLINENEWLINEIn this paper it is introduced the notion of kernel systems by means of the Definition: Let \(G\) be a group. A map \({\mathcal F}\colon M\mapsto M_0={\mathcal F}(M)\) from \(\mathcal{MS}(G)\) to \({\mathcal P}(G)\) such that, for each \(M\in\mathcal{MS}(G)\): (1) \(M_0\) is a normal subgroup of \(M\), (2) for every \(a\in M\setminus M_0\) we have \(C_{M_0}(a)=\{1\}\), and (3) for every \(g\in G\setminus M\) we have \(M_0\cap M_0^g=\{1\}\) is called a kernel system on the group \(G\). -- Let \(G\) be a group and \(\mathcal F\) be a kernel system on \(G\). The pair \((G,{\mathcal F})\) is called a KS-group. If \(\mathcal F\) is clear from (or fixed in) the context then \(G\) is called itself a KS-group. A KS-group is called a CN*-group if it satisfies: (4) \(G=\bigcup_{M\in\mathcal{MS}(G)}M_0\), and: (5) For all \(M\in\mathcal{MS}(G)\), \(M/M_0\) is a nonidentity cyclic group.NEWLINENEWLINENEWLINEThe factorizability hypothesis. (H) \(G\) is a nonsolvable CN*-group, \(p\in\pi(G)\), and \(H\) is a solvable Hall \(p'\)-subgroup of \(G\). Theorem. Under hypothesis (H), one of the following holds: (i) \(p\) is a Fermat prime (\(p=2^{2^m}+1\)) for some \(m\geq 1\) and \(G\simeq\text{SL}_2(\mathbb{F}_{2^{2^m}})\); (ii) \(p=3\) and \(G\simeq\text{SL}_2(\mathbb{F}_8)\). In both cases \(H\) is the normalizer of a Sylow \(2\)-subgroup of \(G\). This theorem generalizes a previous result of the author [\textit{P. Lescot}, Commun. Algebra 18, No.~3, 833-838 (1990; Zbl 0701.20009)].