On subgroups of infinite finitely generated $p$-groups
Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 437-448
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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.
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@article{SM_1987_57_2_a7,
author = {A. V. Rozhkov},
title = {On subgroups of infinite finitely generated $p$-groups},
journal = {Sbornik. Mathematics},
pages = {437--448},
publisher = {mathdoc},
volume = {57},
number = {2},
year = {1987},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1987_57_2_a7/}
}
A. V. Rozhkov. On subgroups of infinite finitely generated $p$-groups. Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 437-448. http://geodesic.mathdoc.fr/item/SM_1987_57_2_a7/