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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