Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T12:52:10.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups with an automorphism cubing many elements

Published online by Cambridge University Press:  09 April 2009

Desmond MacHale
Desmond MacHale
Affiliation:
Department of Mathematics University CollegeCork Ireland.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a group and αn the mapping which takes every element of G to its nth power, where n is an integer. It is well known that if αn is an automorphism then G is Abelian in the cases n = -1,2, and 3. For any other integer n(≠ 0) there exists a non-Abelian group which admits αn as the identity automorphism. Indeed Miller (1929) has shown that if n ≠ 0, ±1, 2, 3 then there exist non-Abelian groups which admit αn as a non-trivial automorphism.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 2 , September 1975 , pp. 253 - 256
Copyright
Copyright © Australian Mathematical Society 1975

References

Joseph, K. S. (1969), Commutativity in non-Abelian groups (Ph. D. thesis, University of California, Los Angeles, 1969).Google Scholar
Liebeck, H. (1973), ‘Groups with an automorphism squaring many elements’, J. Austral. Math. Soc. 16, 33–42.CrossRefGoogle Scholar
Liebeck, H. and MacHale, D. (1973), ‘Groups of odd order with automorphisms inverting many elements’, J. London Math. Soc. (2) 6, 215–223.CrossRefGoogle Scholar
Manning, W. A. (1906), ‘Groups in which a large number of operators may correspond to their inverses’, Trans. Amer. Math. Soc. 7, 233–240.CrossRefGoogle Scholar
Miller, G. A. (1929), ‘Possible α-automorphisms of non-Abelian groups’, Proc. Nat. Acad. Sci. 15, 89–91.CrossRefGoogle ScholarPubMed