Central automorphisms of periodic groups
An automorphism of a group G is called a central automorphism if it commutes with every inner automorphism, or equivalently if it acts trivially on the central factor group G/Z(G). Among the central automorphisms of G there are in particular the power automorphisms, i.e. the automorphisms leaving every subgroup of G invariant. The author studies the action induced by the group \(A_ c(G)\) of all central automorphisms on the group G. His main result is the following: Theorem A. Let G be a group and Z a periodic subgroup of Z(G). If \(A=C_{Aut G}(G/Z)\) and \(B=C_{Aut G}(A)\), then B induces on [G,A] a group of power automorphisms. - In particular, if G is periodic, the centre \(Z(A_ c(G))\) induces on \([G,A_ c(G)]\) a group of power automorphisms. The second main result of the paper is Theorem B. Let G, Z, A and B be as in Theorem A. Then B induces a group of power automorphisms on the torsion subgroup of \(G/C_ G(A)\). - As a consequence of these theorems the author obtains information about periodic groups with nilpotent automorphism groups.
scientific article; zbMATH DE number 4057794
scientific article