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Sylow Theory for a Certain Class of Operator Groups

Published online by Cambridge University Press:  20 November 2018

Christine W. Ayoub
Christine W. Ayoub*
Affiliation:
Pennsylvania State University
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In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 192 - 200
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Ayoub, C. W., A theory of normal chains, Can. J. Math., 4 (1952), 162–188.Google Scholar
2. Ayoub, C. W., On the primary subgroups of a group, Trans. Amer. Math. Soc, 72 (1952), 450–466.Google Scholar
3. Remak, R.. Ueber minimale invariante Untergruppen in der Théorie der endlichen Gruppen, J. reine Angew. Math., 162 (1930), 1–16.Google Scholar