Groups with finite automorphism classes of subgroups
\textit{D. J. S. Robinson} and \textit{J. Wiegold} [ibid. 71, 273-286 (1984; Zbl 0543.20025)] had investigated groups in which the orbits of all subgroups under the automorphism group are boundedly finite, and had shown that this is the case if (and trivially only if) the orbits of only all abelian subgroups are boundedly finite; and had given another, more complicated equivalent condition. In the present paper the authors show that if in a group the orbits of all subgroups are finite, then they are boundedly finite. The proof is elementary, but (perhaps not surprisingly) quite hard. The authors then go on to study groups in which the orbits of subgroups under the endomorphism semigroup are finite; again this is equivalent to them being boundedly finite, and also to the same condition for the abelian subgroups only; and the structure of the group can be quite simply described. Finally they show that a group in which all elements have boundedly finite orbits under the endomorphism semigroup must be finite. Here ``boundedly'' can not be omitted. This generalizes a well-known theorem of \textit{R. Baer} [Trans. Am. Math. Soc. 79, 521-540 (1955; Zbl 0065.010)], which says that a group is finite if its endomorphism semigroup is finite.
scientific article; zbMATH DE number 4065279