The Group of Outer Automorphisms of the Semidirect Product of the Additive Group of a Ring by a Group of Units

Abstract

Let R be a ring with identity, and let U be a subgroup of the unit group of R. Denote by R ⁺ the underlying additive group of R. Consider the action of U on R ⁺ given by left multiplication, and the corresponding semidirect product group R ⁺ ⋊ U. We explicitly compute the automorphism group and the outer automorphism group of R ⁺ ⋊ U, under certain assumptions on R and U. The groups of outer automorphisms of the three euclidean triangle groups, which motivated our investigations, are obtained as particular cases. Many other examples to which our method applies are given.

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Acknowledgment

I wish to thank Gerald Cliff for a useful discussion.

Notes

^#Communicated by M. Dixon.