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Semidirect products of central groups and groups with equal uniformities

Published online by Cambridge University Press:  24 October 2008

R. W. Bagley  and
J. S. Yang
R. W. Bagley
Affiliation:
University of MiamiUniversity of South Carolina, Columbia
J. S. Yang
Affiliation:
University of MiamiUniversity of South Carolina, Columbia
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Extract

Let H and K be topological groups, and let HⓈ K denote the semidirect product determined by a homomorphism (η): H → A(K), where A(K) is the automorphism group of K. In this paper we consider two restricted types of semidirect products. We say that HⓈ K is a semidirect product of type I if η(h) is the identity on Z(K), the centre of K, for each hє H, and of type II if η(H) є I(K), where I(K) is the group of inner automorphisms of K. We obtain conditions under which a type II semidirect product of two groups with equal uniformities has equal uniformities, and conditions under which a type I (hence type II) product of two central groups is central. A group G is central if G/Z(G) is compact, where Z(G) is the centre of G.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 2 , September 1982 , pp. 239 - 241
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

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