Groups with many permutable subgroups
TL;DR: In this paper, the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite.
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Abstract: We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.
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Citations
Sylow permutability in locally finite groups
TL;DR: In this article, the structure of locally finite groups in whose finite subgroups Sylow permutability is a transitive relation is described and a theorem is established which describes the structure.
5
Periodic groups with many permutable subgroups
TL;DR: Curzio and Wiegold as mentioned in this paper showed that if the generators of such a group are periodic then G is locally finite and showed that G is a PH-group if and only if it is centre-by-finite.
2
On the influence of transitively normal subgroups on the structure of some infinite groups
TL;DR: In this paper, the authors describe the structure of a group whose cyclic subgroups (or a part of them) are transitively normal, which is related to the transitivity of normality.
Groups with many elliptic subgroups
Akbar Rhemtulla
- 01 Jan 1989
TL;DR: In this paper, it was shown that a group G has many elliptic pairs of subgroups if every infinite set H, Hj, i ≠ j such that =(HiHj)n for some integer n depending on Hj.
2
References
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The Burnside problem and identities in groups
Sergei Ivanovich Adian,John C. Lennox,James Wiegold +2 more
- 01 Jan 1979
TL;DR: In this article, the authors present a notation for periodic words of rank in a Free Group of Finite Exponents (FGOF) model, where each pair of cyclic subgroups in the FGOF can be represented by a periodic word.
430
A problem of Paul Erdös on groups
TL;DR: In this paper, the authors consider complete subgraphs of a group, or equivalently in sets of elements of the group no two of which commute, and they define complete sub-graphs as a set of complete subsets of the groups of which two vertices do not commute.
Extensions of a problem of Paul Erdös on groups
John C. Lennox,James Wiegold +1 more
TL;DR: In this paper, the main results are as follows: a finitely generated soluble group G is polycyclic if and only if every infinite set of elements of G contains a pair of elements that generate a poly-cyclic subgroup; and the same result with "polycyclic" replaced by "coherent".