Conjugate-Permutable Subgroups

Groups with all cyclic subgroups conjugate-permutable groups

“…This leads to the proof that if G is a group, p an odd prime and Syl p G r CCP, then T i fx e G j x p i 1g is a normal subgroup of G. We also show that if x is a 2-element of a group G and hxi`C ÀP G, then hxi G is a locally ®nite 2-group. The above results lead to some new results and strengthening of results from [1]. For example, it will follow that any torsion group G e CCP is locally ®nite, and if G has ®nite odd exponent, then G is solvable.…”

“…For the reader's convenience and because of its importance for this section we restate Lemma 1.6 of [1]. Lemma 1.6].…”

Permutable subnormal subgroups of finite groups

“…One aim of this article is to give an affirmative answer to this 4 question in the solvable universe. However, we provide a counterexample which 5 shows that the answer is negative for insolvable groups.…”

Product of some elements of finite groups

“…Many new concepts are given from the commutative product of two subgroups in these years. For example, quasinormal subgroup [7], s-quasinormal subgroup [5], conjugate-permutable subgroup [3], etc. However, there are few results for the commutative product of two elements.…”