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On certain subgroups of a join of subnormal subgroups

Published online by Cambridge University Press:  18 May 2009

Howard Smith
Howard Smith
Affiliation:
Department of Pure Mathematics, University College, Cardiff
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1. Introduction: Suppose the group G is generated by subnormal subgroups H and K, and that A, B are normal subgroups of finite index in H, Krespectively. The following question has been asked by J. C. Lennox: Under what circumstances is the subgroupJ = (A, B) subnormal in G? In particular, it is of interest to know when J has finite index in G, for, if this is the case, we may factor out by the normal core of J in G and apply Wielandt's theorem on joins of subnormal subgroups of finite groups [11] to deduce that J is subnormal in G. Here we prove the following result.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 1 , January 1984, pp. 103 - 105
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Amberg, B., Artinian and Noetherian factorized groups, Rend. Sem. Mat. Univ. Padova 55 (1976), 105–122.Google Scholar
2.Baumslag, G., Wreath products and p-groups, Proc. Cambridge Philos. Soc. 55 (1959), 224–231.Google Scholar
3.Lennox, J. C., Segal, D. and Stonehewer, S. E., The lower central series of a join of subnormal subgroups. Math. Z. 154 (1977), 85–89.Google Scholar
4.Lennox, J. C. and Stonehewer, S. E., The join of two subnormal subgroups, J. London Math. Soc. (2) 22 (1980), 460–466.CrossRefGoogle Scholar
5.Robinson, D. J. S., Joins of subnormal subgroups, Illinois J. Math. 9 (1965), 144–168.CrossRefGoogle Scholar
6.Robinson, D. J. S., A property of the lower central series of a group, Math. Z. 107 (1968), 225–231CrossRefGoogle Scholar
7.Robinson, D. J. S., Finiteness conditions and generalised soluble groups, Vol. 1 (Springer, 1972).Google Scholar
8.Roseblade, J. E., The derived series of a join of subnormal subgroups, Math. Z. 117 (1970), 57–69.CrossRefGoogle Scholar
9.Roseblade, J. E. and Stonehewer, S. E., Subjunctive and locally coalescent classes of groups, J. Algebra 8 (1968), 423–435.CrossRefGoogle Scholar
10.Stonehewer, S. E., Nilpotent residuals of subnormal subgroups, Math. Z. 139 (1974), 45–54.CrossRefGoogle Scholar
11.Wielandt, H., Eine Verallgemeinerung der invarienten Untergruppen, Math. Z. 45 (1939), 209–244.Google Scholar