Characterization of groups with finite periodic part
A point of a group G is an element x of finite order such that, (if \(x\neq 1)\) for every finite nontrivial subgroup K of G with \(x\in N_ G(K)\) the normalizer \(N_ G(K)\) has only finitely many finite subgroups containing x. The identity element is a point if and only if the set of torsion elements of G is finite. If \(x\in U\leq G\) is a point of G, it is also a point of U. The existence of points restricts the structure of the group substantially: for instance no group without involutions has an infinite locally finite subgroup containing a point (10.6). Main results: Theorem 1: Let G be a group without involutions and a, b two points of G such that \(<a,g^{-1}bg>\) is finite for almost all conjugates \(g^{-1}bg\). Then at least one of the following holds: (i) The set of torsion elements of G is finite, (ii) There is an infinite set M of finite subgroups of G such that for one of a,b (for a, say) and for all subgroups H in M the subgroup \(<H,a>\) is a finite Frobenius group with noninvariant factor \(\), the set of torsion elements in \(C_ G(H)\) is infinite and a is contained in only finitely many finite subgroups of \(<C_ G(H),a>\). - Theorem 2: A group G without involutions possesses only finitely many torsion elements if and only if there is a pair a, b of points such that, for almost all \(g^{-1}bg\), \(<a,g^{- 1}bg>\) is finite and whenever a or b is contained in a subgroup L with infinitely many torsion elements it is contained in infinitely many finite subgroups of L. - Several examples.
scientific article; zbMATH DE number 3922949