Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T02:44:52.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Endomorphism rings of non-abelian groups

Published online by Cambridge University Press:  17 April 2009

B. C. McQuarrie  and
J. J. Malone
B. C. McQuarrie
Affiliation:
Worcester Polytechnic Institute, Worcester, Massachusetts, USA
J. J. Malone
Affiliation:
Texas A & M University, College Station, Texas, USA.
Rights & Permissions [Opens in a new window]

Abstract

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.

In 1942 F.W. Levi described those groups in which any two inner automorphic images of an arbitrary element commute. Until recently it was not known whether there existed non-abelian groups with the property that any two endomorphic images of an arbitrary element commute. Now, R. Faudree has given examples of finite p–groups having this property. In this paper we give a necessary and sufficient condition that a torsion group which contains no elements of order 2 has this property. The technique of proof involves looking at certain near rings of functions on the group. The inspiration for the theorem comes from the “halving” technique used by B.H. Neumann in 1940.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 3 , December 1970 , pp. 349 - 352
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Chandy, A.J., “Rings generated by inner automorphisms of non-abelian groups”, Doctoral thesis, Boston University, 1965.Google Scholar
[2]Faudree, R.J., “Groups in which each element commutes with its endomorphic images”, Proc. Amer. Math. Soc. (to appear).Google Scholar
[3]Fröhlich, A., “Distributively generated near-rings”, Proc. London Math. Soc. (3) 8 (1958), 76–108.CrossRefGoogle Scholar
[4]Levi, F.W., “Groups in which the commutator operation satisfies certain algebraic conditions”, J. Indian Math. Soc. (N.S.) 6 (1942), 87–97.Google Scholar
[5]Neumann, B.H., “On the commutativity of addition”, J. London Math. Soc. 15 (1940), 203–208.CrossRefGoogle Scholar