Automorphisms of direct products of finite groups II

Abstract

In this paper we first find the automorphism group of the direct product of n copies of an indecomposable non-abelian group. We describe the automorphism group as matrices with entries which are homomorphisms between the n direct factors. We then use this description with a generalization of a result by Bidwell, Curran, and McCaughan on Aut (H ×K), where H and K have no common direct factor, to provide structure and order theorems for an arbitrary direct product.

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“…We consider now the automorphisms of tensor products of Hopf algebras. These are the natural extensions of the corresponding results in group theory [Bidwell et al 2006;Bidwell 2008].…”

“…Thus, we are naturally led to analyze endomorphisms and automorphisms of a tensor product of two Hopf algebras. In [Bidwell et al 2006;Bidwell 2008] an analysis of the automorphisms of direct products of groups was provided. The basic idea is to describe such automorphisms by a matrix of morphisms between the factors.…”

Automorphisms of the Doubles of Purely Non-Abelian Finite Groups

“…It is convenient to think of ψ = (p, u, r, v) as a matrix u r p v , (1.11) with evaluations performed on the right. We may multiply two such matrices together, where, as is standard [6,7], addition is the convolution product-and thus subtraction indicates the antipode-, and multiplication is composition. It is a straightforward exercise to verify that the relations in the Theorem are precisely what is needed to make such a matrix a morphism of Hopf algebras, and for matrix multiplication to correspond to composition of morphisms.…”

“…In both the Corollary and example, the reasons for the existence of such automorphisms are precisely the same as why the description of Aut(G×G) depends on whether G has abelian factors or not. See [6,7] for details.…”

“…Note that the structure, and even membership, of Aut(D(D 2n )) when n ≡ 2 mod 4 is not determined by our results, since then D 2n ∼ = Z 2 ×D n is not purely non-abelian. The completion of the general case, which is analogous to the case for groups [6,7], is left for a future paper. For now, we prove only the following special case.…”