Transitive characteristically simple subgroups of finite quasiprimitive permutation groups (Q2282988)

This paper proves a number of interesting theorems about wreath products in the product action. (Theorem 1.1): Let \(\Gamma\) be a finite set with \(\left\vert \Gamma\right\vert \geq2\) and \(r\geq2\) and let \(W=\mathrm{Sym}(\Gamma)~wr\ S_{r}\) be the wreath product considered as a permutation group on \(\Gamma^{r}\) in the product action; then every transitive nonabelian characteristically simple subgroup of \(W\) is contained in the base group \(\mathrm{Sym}(\Gamma)^{r}\). Recall that a permutation group is quasiprimitive if every nontrivial normal subgroup is transitive. Then (Theorems 1.2 and 1.3): If \(G\) is a finite quasiprimitive permutation group with a nonabelian socle \(S:=\mathrm{soc}(G)\) then every transitive nonabelian characteristically simple subgroup \(H\) of \(G\) is contained in \(S\). Moreover, if \(H\) does not contain a minimal normal subgroup of \(S\) then \(H\) can be described in terms of the O'Nan-Scott classification. (Corollary 1.4): In particular, if \(H\) is regular then \(G\) is of type \(A_{S}\) or \(P_{A}\). (Corollary 1.5): Let \(H\) be a nonabelian characteristically simple group and let \(\Gamma\) be a noncomplete Cayley graph of \(H\). If \(G\) is a quasiprimitive subgroup of \(\mathrm{Aut}(\Gamma)\) with nonabelian socle and \(H\leq G\), then either \(H\) contains a minimal normal subgroup of \(\mathrm{soc}(G)\) or \(G\) has O'Nan-Scott type \(P_{A}.\)