Groups with many permutable subgroups (original) (raw)

Infinite groups with many permutable subgroups

Revista Matematica Iberoamericana, 2008

A subgroup H of a group G is said to be permutable in G, if HK = KH for every subgroup K of G. By a result due to Stonehewer, every permutable subgroup is ascendant although the converse is false. On the other hand, permutability is not a transitive relation. In this paper we study some infinite groups whose ascendant subgroups are permutable. These groups are very close to the groups in which the relation to be a permutable subgroup is transitive.

A Condition in Finite Solvable Groups Related to Cyclic Subgroups

Bulletin of the Australian Mathematical Society, 2010

In this paper, we classify the finite groups belonging to the class of cyclic-transitive groups. A group G is said to be cyclic-transitive if the following condition holds: if x, y, z are nonidentity elements of G such that 〈x,y〉 and 〈y,z〉 are both cyclic, then 〈x,z〉 is also cyclic.

A note on finite 𝒫𝒮𝒯-groups

Journal of Group Theory, 2007

A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.

Periodic groups with many permutable subgroups

Journal of the Australian Mathematical Society, 1992

Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.

Groups with many abelian subgroups

Journal of Algebra, 2011

It is known that a (generalized) soluble group whose proper subgroups are abelian is either abelian or finite, and finite minimal non-abelian groups are classified. Here we describe the structure of groups in which every subgroup of infinite index is abelian.

Some permutability properties related to -hypercentrally embedded subgroups of finite groups

Journal of Algebra, 2003

Let F denote a saturated formation. In this paper we study some properties of F-hypercentrally embedded subgroups, i.e., those subgroups T of a finite group G such that every chief factor of G between its core and its normal closure is F-central in G. We prove that these subgroups form a sublattice of the lattice of all subgroups of G, if F is subgroup-closed. The main result of the paper is the following: if F contains the class of nilpotent groups and G is a soluble group, a subgroup T which permutes with all Sylow subgroups of G is F-hypercentrally embedded in G if and only if T permutes with some F-normalizer of G.

ON THE FINITENESS PROPERTIES OF GROUPS

For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].