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Fibered p-groups. (English) Zbl 0618.20016
Suppose G is a p-group which is not necessarily finite but has non- trivial centre Z. The paper is concerned with the case when G has a partition, that is, a set of at least two subgroups for which any non- identity element of G lies in precisely one of them. The case of exponent p is not of much interest, since the set of all subgroups of order p is such a partition, but it is easy to see that if the exponent of G is greater than p, the subgroup $$H_ p$$ generated by all elements of order greater than p is contained in a member of the partition. Thus $$H_ p<G$$, and conversely, groups for which $$H_ p<G$$ have an obvious partition. Consideration of such groups leads to the definition of a group of type (p,$$\nu)$$ as $$<a,b>$$, when $$| <a>| =p^{\nu}$$, ab$$\neq ba$$, $$b\not\in <a>^ G$$, and $$(b^ ix)=1$$ for $$(i,p)=1$$ and $$x\in <a>^ G$$. Such groups are described in terms of a class of commutative local rings. In the final section, some examples are constructed by means of couplings. The readability of the paper is marred by its unorthodox notation.
Reviewer: N.Blackburn

##### MSC:
 20D15 Finite nilpotent groups, $$p$$-groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D30 Series and lattices of subgroups 16Y30 Near-rings
##### Keywords:
Hughes subgroup; fibered p-groups; partition
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##### References:
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