On complemented nonabelian chief factors of a finite group

A Decomposition Theorem for Group Representations

“…Suppose that there exist some group homomorphisms ρ i : H-groups (B 1 , ρ 1 ) and (B 2 , ρ 2 ) are H-equivalent in the sense of Jiménez-Seral and Lafuente [7].…”

Finite groups with large Chebotarev invariant

“…We say that a group V is a G-group if G acts on V via automorphisms. Following [9], we say that two irreducible G-groups V 1 and V 2 are G-equivalent and we put…”

“…In the abelian case, the converse is true: if V 1 and V 2 are abelian and G-equivalent, then V 1 and V 2 are also G-isomorphic. It is proved (see for example [9,Proposition 1.4]) that two chief factors V 1 and V 2 of G are G-equivalent if and only if either they are G-isomorphic, or there exists a maximal subgroup M of G such that G/ Core G (M ) has two minimal normal subgroups N 1 and N 2 G-isomorphic to V 1 and V 2 respectively. For example, the minimal normal subgroups of a crown-based power L k are all L k -equivalent.…”

An upper bound on the Chebotarev invariant of a finite group

“…In the particular case where V 1 and V 2 are abelian the converse is true: if V 1 and V 2 are abelian and G-equivalent, then V 1 and V 2 are also G-isomorphic. It is proved (see for example [7,Proposition 1.4]) that two chief factors V 1 and V 2 of G are Gequivalent if and only if either they are G-isomorphic between them or there exists a maximal subgroup M of G such that G/ Core G (M ) has two minimal normal subgroups N 1 and N 2 G-isomorphic to V 1 and V 2 respectively. For example, the minimal normal subgroups of a crown-based power L k are all L k -equivalent.…”