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Groups with an automorphism squaring many elements

Published online by Cambridge University Press:  09 April 2009

Hans Liebeck
Hans Liebeck
Affiliation:
Department of Mathematics University of Keele Staffs. ST5 5BG, England
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A universal power automorphism (Cooper [1]) of a group is an automorphism mapping every element x to a power xn for some fixed integer n. It is long known that a group admitting such an automorphism with n= −1, 2 or 3 must be Abelian. Miller [5] showed that for every other non-zero integral value of n there exist non-Abelian groups admitting a non-trivial universal power automorphism x→xn.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 1 , August 1973 , pp. 33 - 42
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Cooper, C. D. H., ‘Power automorphisms of a group’. Math. Z. 107 (1968) 335–356.CrossRefGoogle Scholar
[2]Joseph, K. S., Commutativity in non-A belian groups (Ph.D. thesis, University of California, Los Angeles, 1969).Google Scholar
[3]Liebeck, H. and MacHale, D., ‘Groups with automorphisms inverting most elements.’ Math. Z. 124 (1972), 51–63.CrossRefGoogle Scholar
[4]Liebeck, H. and MacHale, D. ‘Odd order groups with automorphisms inverting many elements.’ J. London. Math. Soc. (2) 6 (1973), 215–223.CrossRefGoogle Scholar
[5]Miller, G. A., ‘Possible α-automorphisms of non-Abelian groups’, Proc. Nat. Acad. Sci. 15 (1929), 89–91.CrossRefGoogle ScholarPubMed