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Constructing an Automorphism From an Anti-Automorphism

Published online by Cambridge University Press:  20 November 2018

Christine Ayoub
Christine Ayoub*
Affiliation:
Pennsylvania State University
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We consider the following problem: Let G be a group with distinct automorphisms β and σ and an anti-automorphism α such that

What can be said about G?

If σ = α, σ is both an automorphism and an anti-automorphism so that G is abelian. Hence we assume that σ ≠ α. In this case, we show that G is non-abelian, but has an abelian subgroup of index 2. Conversely, for such a group G there always exist distinct automorphisms β and σ and an anti-automorphism α such that (1) holds.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 3 , August 1968 , pp. 367 - 370
Copyright
Copyright © Canadian Mathematical Society 1968