Subnormality, Ascendancy, and the Minimal Condition on Subgroups
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 41, 58-78 (1976) Subnormality, Ascendancy, and the Minimal Condition on Subgroups B. HARTLEY Mathematics Institute, University of Warwick, Coventry, England AND T. A. PENG Department of Mathematics, University of Singapore Communicated by J. E. Roseblade Received June 19, 1974 1. INTRODUCTION In his recent papers [l 1, 121, Wielandt has given a very elegant treatment of subnormality in finite groups, leading to a number of criteria for a subgroup of a finite group to be subnormal. The well-known fact, first proved by Baer, that every left Engel element of a finite group lies in the Fitting subgroup, appears as a special case of Wielandt’s results; this is because an element of finite group lies in the Fitting subgroup if and only if the cyclic subgroup it generates is subnormal. Wielandt has also extended some of his results to groups satisfying the maximal condition on subgroups; our concern here is to extend them to groups satisfying Min, the minimal condition on subgroups. As a by-product, we obtain some alternative proofs of known results about left Engel elements of groups satisfying Min. It is appropriate in infinite groups to consider not only subnormal sub- groups, but also subgroups belonging to series of a group other than finite ones. In particular we shall consider ascendant subgroups. A subgroup H of a group G is ascendant in G (written H asc G) if there is an ordinal p and subgroups {H,: 01 < p} of G such that H = HO, H, Q H,+,(a: < p), f-L = UacuHa if p < p is a limit ordinal, and H, = G.
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