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Groups with Metacyclic Sylow 2-Subgroups

Published online by Cambridge University Press:  20 November 2018

A. R. Camina  and
T. M. Gagen
A. R. Camina
Affiliation:
University of East Anglia, Norwich, England
T. M. Gagen
Affiliation:
University of Sydney, Sydney, Australia
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A group S is said to be metacyclic if it contains a normal cyclic subgroup N such that S/N is cyclic. In this note the following theorem is proved.

THEOREM. Let G be a group, S a metacyclic Sylow 2-subgroup of G. If S has a cyclic normal subgroup N such that S/N is cyclic of order greater than 2, then G is soluble.

Remark. We show that such a group G contains a 2-nilpotent normal subgroup of index a divisor of 6. The solubility of these groups requires the solubility of groups of odd order unavoidably.

Notation. All groups considered will be finite. Let G be a group, S a subset of G, A and B subgroups of G, N a normal subgroup of G.

〈S〉: the subgroup of G generated by S.

NG(S): the normalizer of S in G.

CG(S): the centralizer of S in G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1234 - 1237
Copyright
Copyright © Canadian Mathematical Society 1969

References

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