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Supersoluble and Related Cofinitary Groups
Author(s) -
Wehrfritz B. A. F.
Publication year - 1995
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19951760122
Subject(s) - mathematics , vector space , identity (music) , combinatorics , group (periodic table) , element (criminal law) , division ring , zero (linguistics) , space (punctuation) , finite group , normal subgroup , fixed point , pure mathematics , division (mathematics) , mathematical analysis , arithmetic , physics , linguistics , philosophy , quantum mechanics , political science , acoustics , law
Let D be a division ring (possibly commutative) and V an infinite‐dimensional left vector space over D . We consider irreducible subgroups G of GL( V ) containing an element whose fixed‐point set in V is non‐zero but finite dimensional (over D ). We then derive conclusions about cofinitary groups, an element of GL( V ) being cofinitary if its fixed‐point set is finite dimensional and a subgroup G of GL( V ) being cofinitary if all its non‐identity elements are confinitary. In particular we show that an irreducible cofinitary subgroup G of GL( V ) is usually imprimitive if G is supersoluble and is frequently imprimitive if G is hypercyclic. The latter includes the case where G is hypercentral, which apparently is also new.