Automorphisms and stem extensions (Q760503)

A stem extension of a finite group G is a short exact sequence \(e: O\to A\to E\to G\to O\) of finite groups such that \(A\leq Z(E)\cap E'\). A stem cover (or representation group) of G is any stem extension \(e: O\to A\to E\to G\to O\) of G such that \(| A| =| H^ 2(G,{\mathbb{C}}^ x)|\). A finite group G has property \({\mathcal L}\) if for every \(\alpha\in aut(G)\), there is a stem cover \(e: O\to A\to E\to G\to O\) and an automorphism \(\beta\in aut(E)\) that ''induces'' \(\alpha\) (i.e. \(\beta\) leaves A invariant). Section 1 of this paper studies the natural action of aut(G) on the stem extensions of G and deduces connections between this aut(G)-set and the property \({\mathcal L}\). More machinery is developed in Sections 2 and 3. These results are used in Sections 4-5 to obtain deeper results concerning the action of aut(G) on the stem extensions of G. An example of such a result is Corollary 4.5: If every Sylow subgroup of G has property \({\mathcal L}\), then G also has property \({\mathcal L}.\) Sections 6-7 of this paper are devoted to proving Theorem 7.1: Suppose that G is elementary abelian of order \(2^ r\). Then G has property \({\mathcal L}\) if and only if \(r\leq 2\).