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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2011, Volume 17, Number 4, Pages 53–65
(Mi timm749)
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On finite Alperin p-groups with homocyclic commutator subgroup
B. M. Veretennikov Ural Federal University
Abstract:
We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number of generators d(G) of a finite Alperin p-group G is n≥3, then d(G′)≤C2n for p≠3 and d(G′)≤C2n+C3n for p=3. The first section of the paper deals with finite Alperin p-groups G with d(G)≥3 and p≠3 that have a homocyclic commutator subgroup of rank C2n. In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with a homocyclic commutator subgroup G′ of rank C2n+C3n, then G′ is an elementary abelian group.
Keywords:
p-group, Alperin group, commutator subgroup, definition of group by means of generators and defining relations.
Received: 05.02.2011
Citation:
B. M. Veretennikov, “On finite Alperin p-groups with homocyclic commutator subgroup”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 4, 2011, 53–65; Proc. Steklov Inst. Math. (Suppl.), 279, suppl. 1 (2012), 139–151
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https://www.mathnet.ru/eng/timm749 https://www.mathnet.ru/eng/timm/v17/i4/p53
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Abstract page: | 348 | Full-text PDF : | 83 | References: | 68 | First page: | 1 |
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