Compactness conditions for groups of automorphisms of topological groups

It is proved that if G is a compact, totally disconnected Abelian group and Aut G is its group of topological automorphisms (with the natural topology), then the following conditions are equivalent: (a) Aut G is compact; (b) Aut G is locally compact; (c) Aut G has small invariant neighborhoods of the identity; (d) Aut G is an-group; (e) the factor group of Aut G by its center is compact; (f) the closure of the commutator subgroup of Aut G is compact; (g), where F_p is a finite p-group, Z_p is the additive group of p-adic integers, and n_p < .Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 735–743, May, 1976.In conclusion, the author thanks V. P. Platonov for his constant attention to this paper.