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Disjoint conjugates of cyclic subgroups of finite groups

Published online by Cambridge University Press:  20 January 2009

Thomas J. Laffey
Thomas J. Laffey
Affiliation:
Mathematics Department, University College, Dublin 4
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In an earlier paper (2) we considered the following question “If S is a cyclic subgroup of a finite group G and S ∩ F(G) = 1, where F(G) is the Fitting subgroup of G, does there necessarily exist a conjugate Sx of S in G with S ∩ Sx = l?” and we gave an affirmative answer for G simple or soluble. In this paper we answer the question affirmatively in general (in fact we prove a somewhat stronger result (Theorem 3)). We give an example of a group G with a cyclic subgroup S such that (i) no nontrivial subgroup of S is normal in G and (ii) no x exists for which S ∩ Sx = 1.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 3 , March 1977 , pp. 229 - 232
Copyright
Copyright © Edinburgh Mathematical Society 1977

References

REFERENCES

(1) Barriga, O. E., On a problem about cyclic subgroups of finite groups, Proc. Edinburgh Math. Soc. 20 (1977), 225–228.CrossRefGoogle Scholar
(2) Laffey, T. J., A problem on cyclic subgroups of finite groups, Proc. Edinburgh Math. Soc. 18 (1973), 247–249.CrossRefGoogle Scholar