On \(\{2,3\}\)-groups without elements of order 6. (Q2352691)

The aim of the paper under review is the study of \(\{2,3\}\)-groups that do not contain elements of order \(6\). In 1937 \textit{B. H. Neumann} [J. Lond. Math. Soc. 12, 195-198 (1937; Zbl 0016.39303; JFM 63.0066.03)] proved that a group of exponent \(6\) without elements of order \(6\) is locally finite (the general Burnside problem for groups of exponent \(6\) was solved by \textit{M. Hall} in 1956 [Ill. J. Math. 2, 764-786 (1958; Zbl 0083.24801)]). The local finiteness of groups of exponent \(12\) without elements of order \(6\) was proved by \textit{I. N. Sanov} ([Leningr. Gos. Univ., Uch. Zap., Ser. Mat. Nauk 10(55), 166-170 (1940; Zbl 0061.02506)], see also [\textit{D. V. Lytkina}, Sib. Mat. Zh. 48, No. 2, 353-358 (2007); translation in Sib. Math. J. 48, No. 2, 283-287 (2007; Zbl 1154.20036)]). Later, with the additional hypothesis \(O_2(G)\neq G\neq O_3(G)\), similar results were obtained for groups of exponent \(24\) ([\textit{V. D. Mazurov}, Algebra Logic 49, No. 6, 515-525 (2011); translation from Algebra Logika 49, No. 6, 766-781 (2010; Zbl 1225.20034)]), \(36\) ([\textit{E. Jabara} and \textit{D. V. Lytkina}, Sib. Math. J. 54, No. 1, 29-32 (2013); translation from Sib. Mat. Zh. 54, No. 1, 44-48 (2013; Zbl 1285.20036)]) and \(72\) ([\textit{E. Jabara} and the authors, J. Group Theory 17, No. 6, 947-955 (2014; Zbl 1323.20032)]). The main result proved in this paper is: Let \(G\) be a \(\{2,3\}\)-group with \(O_2(G)\neq G\neq O_3(G)\) and without elements of order \(6\). Then the following conditions are equivalent: (1) \(O_2(G)O_3(G)\neq 1\); (2) there exists a nonempty \(G\)-invariant subset \(\Lambda\) of elements of order \(3\) of \(G\) such that \(\langle x,y\rangle\) is finite for all \(x,y\in\Lambda\); (3) \(G\) satisfies one of the conditions: (a) \(G=O_3(G)\cdot T\), where \(O_3(G)\) is a nontrivial abelian \(3\)-group and \(T\) is a locally cyclic or locally quaternion \(2\)-group acting freely on \(O_3(G)\), in particular \(G\) is a locally finite Frobenius group; (b) \(G=O_2(G)\cdot R\), where \(O_2(G)\) is a nontrivial \(2\)-group of nilpotency class at most \(2\) and \(R\) is a \(3\)-group with a unique subgroup of order \(3\) acting freely on \(O_2(G)\), in particular \(G\) is a Frobenius group; (c) \(G=O_2(G)\cdot(R\cdot\langle t\rangle)\), where \(O_2(G)\) is a nontrivial \(2\)-group of nilpotency class at most \(2\), \(R\) is a locally cyclic \(3\)-group acting freely on \(O_2(G)\), and \(t\) is an element of order 2 which inverts every element of \(R\) under conjugation, in particular \(O_2(G)\cdot R\) is a locally finite Frobenius group. The reviewer remarks that the local finiteness of groups of exponent \(12\) with elements of order \(6\) but without elements of order \(12\) is proved by \textit{A. S. Mamontov}, [in Sib. Math. J. 54, No. 1, 114-118 (2013); translation from Sib. Mat. Zh. 54, No. 1, 150-156 (2013; Zbl 1273.20032)].