The complete reducibility of some $GF(2)A_7$-modules

A. S. Kondrat'ev   ; I. V. Khramtsov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 139-143

Résumé

It is proved that, if $G$ is a finite group with a nontrivial normal $2$-subgroup $Q$ such that $G/Q\cong A_7$ and an element of order $5$ from $G$ acts without fixed points on $Q$, then the extension of $G$ by $Q$ is splittable, $Q$ is an elementary abelian group, and $Q$ is the direct product of minimal normal subgroups of $G$ each of which is isomorphic, as a $G/Q$-module, to one of the two $4$-dimensional irreducible $GF(2)A_7$-modules that are conjugate with respect to an outer automorphism of the group $A_7$.

Détail

Keywords: finite group, $GF(2)A_7$-module, completely reducible representation, prime graph.

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