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A subgroup X of a group G is said to be pronormal if the subgroups X and (Formula presented.) are conjugate in (Formula presented.) for all (Formula presented.). Moreover, in analogy with metahamiltonian groups (i.e. groups in which every non-abelian subgroup is normal), a group in which every non-abelian subgroup is pronormal is called prohamiltonian. In this article we will determine those finite simple groups which are prohamiltonian. It will easily follow that the only prohamiltonian locally finite simple groups are the finite ones.

A group G is said to be an FC-group if every conjugacy class of G has finite order and is said to be a T-group if every subnormal subgroup is normal in G. In this paper we give a characterization of groups that are both FC-groups and T-groups and go further in the study of FC-groups with few subnormal non-normal subgroups in the sense of the chain conditions. The structure of FC-groups satisfying the maximal, the minimal and the double chain condition on subnormal non-normal subgroups will be described.

Thepronormof a groupGis the set$P(G)$of all elements$g\in G$such thatXand$X^g$are conjugate in${\langle {X,X^g}\rangle }$for every subgroupXofG. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble groupGwhose pronorm contains a subgroup of finite index.

If θ is a subgroup property, a group G is said to satisfy the double chain condition on θ-subgroups if it admits no infinite double sequences ⋯ < X - n < ⋯ < X - 1 < X 0 < X 1 < ⋯ < X n < ⋯ \cdots consisting of θ-subgroups. The structure of generalised radical groups satisfying the double chain condition on non-subnormal subgroups is described.

An endomorphism [Formula: see text] of a group [Formula: see text] is called a cyclic endomorphism if the subgroup [Formula: see text] is cyclic for all elements [Formula: see text] of [Formula: see text]. It can be proved that every cyclic endomorphism is normal, i.e. it commutes with every inner automorphism of [Formula: see text] (see [F. de Giovanni, M. L. Newell and A. Russo, On a class of normal endomorphisms of groups, J. Algebra its Appl. 13 (2014) 6pp.]). In this paper, some further properties of cyclic endomorphisms will be pointed out. Moreover, the structure of a group [Formula: see text] in which the group [Formula: see text] of cyclic automorphisms has finite index in [Formula: see text] will be investigated.

If [Formula: see text] is a subgroup property, a group [Formula: see text] is said to satisfy the double chain condition on [Formula: see text]-subgroups if it admits no infinite double sequences [Formula: see text] consisting of [Formula: see text]-subgroups. The structure of generalized radical groups satisfying the double chain condition on non-abelian subgroups is described.

If $\unicode[STIX]{x1D703}$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $\unicode[STIX]{x1D703}$-subgroups if it admits no infinite double sequences $$\begin{eqnarray}\cdots consisting of $\unicode[STIX]{x1D703}$-subgroups. We describe the structure of generalised radical groups satisfying the double chain condition on abelian subgroups.

The aim of this article is to prove the following theorem. Let G be any infinite simple locally finite group. Then, either G is isomorphic to $$\mathrm{{PSL}}(2,F)$$ , where F is an infinite locally finite field, or G contains a subgroup which is the direct product of an infinite abelian subgroup of prime exponent p and a finite non-abelian p-subgroup.

If $$\theta $$ is a subgroup property, a group G is said to satisfy the double chain condition on $$\theta $$ -subgroups if it admits no infinite double sequences $$\begin{aligned} \cdots