Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T01:28:41.201Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Brauer, R., Some applications of the theory of blocks of characters of finite groups. II,]. Algebra 1 (1964), 307–334 Google Scholar
2. Glauberman, G., Central elements in core free groups, J. Algebra 4 (1966), 403–420.Google Scholar
3. Hall, M., Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
4. Hall, P. and Higman, G., The p-length of a p-soluble group and reduction theorems for Burnside's problem, Proc. London Math. Soc. 7 (1956), 1–42.Google Scholar
5. Suzuki, M., A characterization of simple groups LF(2, p), J. Fac. Sci. Univ. Tokyo (Sect. I) 6 (1951), 259–293.Google Scholar
6. Thompson, J. G., Non-solvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437.Google Scholar
7. Zassenhaus, H. J., The theory of groups (Chelsea, New York, 1958).Google Scholar