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The subnormal structure of some classes of coluble groups

Published online by Cambridge University Press:  09 April 2009

D. McDougall
D. McDougall
Affiliation:
Institute of Advanced Studies, Australian National UniversityCanberra A.C.T. 2600
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Finite groups in which normality is transitive have been studied by Best and Taussky, [1], Gaschütz, [3], and Zacher [16]. Infinite soluble groups in which normality is transitive have been studied by Robinson in [9]. A subgroup H of a group G is subnormal in G if H can be connected to G by a chain of r subgroups, in which each is normal in its successor, where r is a non-negative integer. The least such r is called the subnormal index of H in G (or the defect of H in G). Then groups in which normality is transitive are precisely those in which every subnormal subgroup has subnormal index at most one. Thus the structure of soluble groups in which every subnormal subgroup has subnormal index at most n (such a group is said to have bounded subnormal indices) has been dealt with by Robinson in [9] for the case where n is one. However Theorem D of [12] states that a soluble group of derived length n can be embedded in a soluble group in which the subnormal indices are at most n. Therefore we must impose further conditions on the groups if we hope to obtain any worthwhile results for the above problem with n greater than one.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 3 , May 1972 , pp. 365 - 377
Copyright
Copyright © Australian Mathematical Society 1972

References

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