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Semidirect products of central groups and groups with equal uniformities

  • R. W. Bagley (a1) and J. S. Yang (a1)

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Let H and K be topological groups, and let HK denote the semidirect product determined by a homomorphism (η): HA(K), where A(K) is the automorphism group of K. In this paper we consider two restricted types of semidirect products. We say that HK 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.

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(1)Bagley, R. W.Semi-direct products of topological groups. Math. Japonica 22 (1977), 2526.
(2)Bagley, R. W. and Lau, K. K.Semi-direct products of topological groups with equal uniformities. Proc. Amer. Math. Soc. 29 (1971), 179182.
(3)Bagley, R. W. and Wu, T. S.Topological groups with equal left and right uniformities. Proc. Amer. Math. Soc. 18 (1967), 142147.
(4)Dugundji, J.Topology (Allyn and Bacon, Boston, 1966).
(5)Grosser, S. and Moskowitz, M.On central topological groups. Trans. Amer. Math. Soc. 127 (1967), 317340.
(6)Scarborough, C. T.Closed graphs and closed projections. Proc. Amer. Math. Soc. 20 (1969), 466470.

Semidirect products of central groups and groups with equal uniformities

  • R. W. Bagley (a1) and J. S. Yang (a1)

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