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Subsocles Supporting Isotype and Balanced Subgroups

Published online by Cambridge University Press:  20 November 2018

Paul Hill  and
Charles Megibben
Paul Hill
Affiliation:
Department of Algebra Combinatorics and Analysis Auburn University Auburn, Alabama 36849 U.S.A.
Charles Megibben
Affiliation:
Department of Mathematics Vanderbilt University Nashville, Tennessee 37240 U.S.A.
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Abstract

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We identify a condition, which we refer to as cohesiveness, on a subgroup S of the socle G[p] — {x ∊ G : px = 0} of an abelian p-group G which is necessary for S to be the socle of an isotype subgroup of G. It is shown, when S is countable, that this condition is both necessary and sufficient. A further restriction, definable in terms of the coset valuation on G/S, leads to the notion of S being completely cohesive in G. When S is countable, this latter condition is both necessary and sufficient for S to serve as the socle of a balanced subgroup of G. Also noteworthy is the fact that if H and K are, respectively, balanced and isotype subgroups of G with H[p] = K[p], then K is necessarily balanced in G.

Keywords

20K1020K27
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 4 , 01 December 1993 , pp. 419 - 425
Copyright
Copyright © Canadian Mathematical Society 1993

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