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Journal of the London Mathematical Society 1974 s2-9(1):30-34; doi:10.1112/jlms/s2-9.1.30
© 1974 by London Mathematical Society
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© Oxford University Press

Almost Locally Invariant Topological Groups

Edward T. Ordman and Sidney A. Morris

Department of Mathematics, University of New South Wales Kensington, N.S.W., Australia 2033 University of Kentucky Lexington, Kentucky 40506, U.S.A.
Department of Mathematics, University of New South Wales Kensington, N.S.W., Australia 2033

Maximally almost periodic groups and locally invariant groups have been studied extensively in the literature. Maximally almost periodic groups are those admitting continuous monomorphisms into compact groups; locally invariant groups are those in which every neighbourhood of the identity contains a neighbourhood invariant under inner automorphisms. in this paper a study is made of almost locally invariant groups, which are groups admitting continuous monomorphisms into locally invariant groups. This class includes all maximally almost periodic groups and all locally invariant groups, but there exist locally compact almost locally invariant groups which are neither locally invariant nor maximally almost periodic. a locally compact almost locally invariant group which is connected or locally connected is locally invariant. The class of almost locally invariant groups is closed under passage to subgroups, direct products, and free products, but not quotients.


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