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On cyclic subgroups of finite groups

Published online by Cambridge University Press:  20 January 2009

U. Dempwolff  and
S. K. Wong
U. Dempwolff
Affiliation:
Universität KaiserslauternFB MathematikPostfad 3049D-675 Kaiserslautern
S. K. Wong
Affiliation:
The Ohio State UniversityDepartment of Mathematics231 W. 18th Ave.Columbus, Ohio 43210, U.S.A.
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In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:

Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 19 - 20
Copyright
Copyright © Edinburgh Mathematical Society 1982

References

REFERENCES

1.Bender, H., On groups with Abelian Sylow 2-subgroups, Math. Z. 117 (1970), 164–176.CrossRefGoogle Scholar
2.Gorenstein, D., Finite groups (Harper and Row Publishers, New York, 1968).Google Scholar
3.Laffey, T. J., Disjoint conjugates of cyclic subgroups of finite groups, Proc. Edinburgh Math. Soc. 20 (1976–1977), 229–232.CrossRefGoogle Scholar