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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 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 < · · · consisting of θ -subgroups. The structure of generalized soluble groups satisfying the double chain condition on subnormal subgroups is described.

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