On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup |
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Authors: | B M Veretennikov |
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Institution: | 1. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia
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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 d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C n 2 for p≠ 3 and d(G′) ≤ C n 2 + C n 3 for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C n 2 . 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 homocyclic commutator subgroup G- of rank C n 2 + C n 3 , then G″ is an elementary abelian group. |
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