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Permutable subgroups of some finite p-groups

Published online by Cambridge University Press:  09 April 2009

S. E. Stonehewer
S. E. Stonehewer
Affiliation:
Mathematics Institute University of WarwickCoventry, England
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A subgroup H of a group G is said to be a permutablesubgroup of G if HK = KH 〈H, K〉 for all subgroups K of G. It is known that a core-free permutable subgroup H of a finite group G is always nilpotent [5]; and even when G is not finite, H is always a subdirect product of finite nilpotent groups [11]. Thus nilpotency is a measure of the extent to which a permutable subgroup differs from being normal. Examples of non-abelian, core-free, permutable subgroups are rare and difficult to construct. The first, due to Thompson [12], had class 2. Further examples of class 2 appeared in [8]. More recently Bradway, Gross and Scott [1] have constructed corresponding to each positive integer c and each prime p ≥ c, a finite p-group possessing a core-free permutable subgroup of class c. In [3] Gross succeeded in dispensing with the requirement p ≥ c.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 1 , August 1973 , pp. 90 - 97
Copyright
Copyright © Australian Mathematical Society 1973

References

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