On group automorphisms which centralize the factor group by an Abelian normal subgroup (Q1063110)

Let \(A\) be a normal Abelian subgroup of the group \(G\), denote by \(R\) the ring consisting of the cocycles of \(G\) into \(A\) with point-wise addition and composition as multiplication, let \(Q(R)\) denote the circle group of \(R\) (the elements of \(R\) with the circle operation, \(a\circ b=a+b+ab\)), and let \(D\) be the group of automorphisms of \(G\) which normalize \(A\). Then (Lemma 3.1) the group \(C\) of automorphisms of \(G\) which fix each coset of \(A\) is \(D\)-isomorphic with \(Q(R)\). This connection between automorphism groups and circle groups of rings enables the author to show, for example, that the group of automorphisms of \(G\) which centralize \(G/A\) when \(A\) is finite has the same structure as the automorphism group of a finite Abelian group (as determined by K. Shoda in his well-known paper). The author's main result (Theorem 3.5) includes Shoda's result and Fitting's results on the group of central automorphisms as special cases, a good indication of the cleverness of the author's approach.