On torsion in groups whose automorphism groups have finite rank (Q1102374)

This paper studies the consequences for the elements of finite order in a soluble-by-finite group of imposing certain finiteness of rank conditions on its automorphism group. A group has finite abelian subgroup rank if every abelian subgroup A has finite torsion-free rank \(r_ 0(A)\) and finite p-rank \(r_ p(A)\) for all primes p. A group has finite abelian total rank if the total rank \(r_ 0(A)+\sum_{p}r_ p(A)\) of each abelian subgroup A is finite. The main results of this paper concern soluble groups. Theorem 1: Let G be a soluble-by-finite group. (a) If Aut G has finite abelian subgroup rank, then each Sylow subgroup of G is a Chernikov group. (b) If Aut G satisfies the maximal condition on abelian subgroups then the Sylow subgroups of G are finite. (c) If Aut G satisfies the minimal condition on abelian subgroups then Aut G is finite and the elements of finite order form a finite subgroup of G. Theorem 2: Let G be a soluble-by-finite group which is also torsion-by- nilpotent. (a) If Aut G has finite abelian subgroup rank, then the Sylow subgroups of G are finite. (b) If Aut G has finite abelian total rank, then the elements of finite order form a finite subgroup of G. - The authors construct examples to illustrate the limitations to Theorem 1 and 2.