Endimioni, GĂ©rard Groups in which certain equations have many solutions. (English) Zbl 1072.20035 Rend. Semin. Mat. Univ. Padova 106, 77-82 (2001). For a group word \(w(x_1,\dots,x_n)\) and a cardinal \(\kappa\), let \({\mathcal V}_\kappa(w)\) be the class of all groups \(G\) such that every subset of \(G\) of cardinality \(\kappa\) contains \(n\) distinct elements \(g_1,\dots,g_n\) with \(w(g_1,\dots,g_n)=1\). For example, if \(w\) is \([x_1,x_2]\) then \({\mathcal V}_\omega(w)\) is the class of all groups \(G\) without infinite sets of pairwise non-commuting elements; as B. H. Neumann proved [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)] they are exactly the central-by-finite groups. The author considers the word \(w=x_1^{\alpha_1}\cdots x_n^{\alpha_n}\), where \(\alpha_1,\dots,\alpha_n\) are nonzero integers. The main result: for an infinite group \(G\), the following are equivalent: (1) \(G\in{\mathcal V}_\omega(w)\); (2) \(G\in{\mathcal V}_m(w)\), for some \(m\in\omega\); (3) \(G\in{\mathcal B}_{\alpha_1+\cdots+\alpha_n}\cap\mathcal{FB}_\alpha\), where \(\alpha\) is the greatest common divisor of \(\alpha_1,\dots,\alpha_n\). Here \({\mathcal B}_\alpha\) is the variety of groups satisfying \(x^\alpha=1\), and \(\mathcal F\) is the class of finite groups. Reviewer: Oleg V. Belegradek (Istanbul) Cited in 1 ReviewCited in 1 Document MSC: 20E10 Quasivarieties and varieties of groups 20F05 Generators, relations, and presentations of groups 20F12 Commutator calculus Keywords:varieties of groups; group words PDF BibTeX XML Cite \textit{G. Endimioni}, Rend. Semin. Mat. Univ. Padova 106, 77--82 (2001; Zbl 1072.20035) Full Text: Numdam EuDML References: [1] A. Abdollahi , Some Engel conditions on infinite subsets of certain groups , Bull. Austral. Math. Soc. , 62 ( 2000 ), pp. 141 - 148 . MR 1775895 | Zbl 0964.20019 · Zbl 0964.20019 · doi:10.1017/S0004972700018554 [2] A. Abdollahi - B. TAERI, A condation on a certain variety of groups , Rend. Sem. Mat. Univ. Padova , 104 ( 2000 ), pp. 129 - 134 . Numdam | MR 1809354 | Zbl 1013.20021 · Zbl 1013.20021 · numdam:RSMUP_2000__104__129_0 · eudml:108531 [3] G. Endimioni , On a combinatortal problem in varieties of groups , Comm. Algebra , 23 ( 1995 ), pp. 5297 - 5307 . MR 1363602 | Zbl 0859.20021 · Zbl 0859.20021 · doi:10.1080/00927879508825531 [4] P. Longobardi - M. MAJ - A. H. RHEMTULLA, Infinite groups in a gaven vanety and Ramsey’s theorem , Comm. Algebra , 20 ( 1992 ), pp. 127 - 139 . MR 1145329 | Zbl 0751.20020 · Zbl 0751.20020 · doi:10.1080/00927879208824335 [5] P. Longobardi - M. MAJ, Finitely generated soluble groups with an Engel condition on infinite subsets , Rend. Sem. Mat. Univ. Padova , 89 ( 1993 ), pp. 97 - 102 . Numdam | MR 1229046 | Zbl 0797.20031 · Zbl 0797.20031 · numdam:RSMUP_1993__89__97_0 · eudml:108296 [6] B.H. Neumann , A problem of Paul Erdos on groups , J. Austral. Math. Soc. , 21 ( 1976 ), pp. 467 - 472 . MR 419283 | Zbl 0333.05110 · Zbl 0333.05110 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.