The embedding of a cyclic permutable subgroup in a finite group.
A subgroup \(A\) of a group \(G\) is said to be permutable if \(AX=XA\) for all subgroups \(X\) of \(G\). In an ealier paper [J. Aust. Math. Soc. 71, No. 2, 169-176 (2001; Zbl 1001.20010)], the authors proved that if \(A\) is a cyclic permutable subgroup of odd order of a finite group \(G\), then (1) \([A,G]\) is Abelian; (2) \(A\) acts by conjugation on \([A,G]\) as a group of power automorphisms. The situation turns out to be much more complicated when \(A\) has even order. Results are obtained under the additional condition (*): each cyclic subgroup \(X\) of \(G\) is permutable in \(A_2X\), where \(A_2\) is the \(2\)-component of \(A\). Theorem A: Let \(A\) be a cyclic permutable subgroup of a finite group \(G\) satisfying (*), and let \(N\) be the derived subgroup of \([A,G]\). Then (i) \(N\) has order at most \(2\) and lies in \(A\); and (ii) \(A\) acts on \([A,G]/N\) as a group of universal power automorphisms [a power automorphism is universal if it maps all elements to the same power]. Theorem B: Let \(A\) be a cyclic permutable subgroup of a finite group \(G\) satisfying (*). Then \([A,G]\) is Abelian if and only if \(A\) acts on \([A,G]\) as a group of power automorphisms.
scientific article; zbMATH DE number 1993621