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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb. (N.S.), 1986 Volume 129(171), Number 3, Pages 422–433 (Mi sm1837)

This article is cited in 3 papers

On subgroups of infinite finitely generated $p$-groups

A. V. Rozhkov


Abstract: A complete description is given of the abelian subgroups of an arbitrary Alëshin $p$-group (RZh. Mat., 1972, 8A271; 1983, 1A189K § 23). They are precisely the direct sums of at most countably many cyclic $p$-groups (Theorem 2). For $p>2$, a description is given of the subgroups acting nontrivially on sequences with given initial segment only (Theorems 1 and 1$'$), whence it follows in particular that nontrivial normal subgroups are of finite index (Algebra i Logika, 1983, V. 22, № 5, P. 584–589). The centralizer of every element of every Alëshin group is infinite (Theorem 3). An infinite subgroup generated by two conjugate elements of prime order is constructed in an Alëshin group, thus answering in the negative problem 6.58a) of the Kourovka Notebook (RZh. Mat., 1984, 2A156K), due to V. P. Shunkov.
Figures: 1.
Bibliography: 9 titles.

UDC: 512.5

MSC: Primary 20F50, 20E07; Secondary 20F10

Received: 14.06.1983 and 15.01.1985


 English version:
Mathematics of the USSR-Sbornik, 1987, 57:2, 437–448

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