Bounding the exponent of a finite group with automorphisms (Q1284115)

The authors consider the following situation: (a) \(A\) is a noncyclic Abelian group of automorphisms of a finite group \(G\), \((|A|,|G|)=1\), (b) \(q\) is the smallest prime such that the Sylow-\(q\)-subgroup of \(A\) is noncyclic, (c) the least common multiple of the exponents of the centralizers \(C_G(a)\), with \(a\in A\) and non-identity, is \(n\). They show that the exponent of \(G\) is bounded in terms of \(q\) and \(n\) in this situation (Theorem 1). As a consequence they show that a torsion compact group possessing an elementary Abelian subgroup with finite centralizer has bounded exponent. The proof is via Lie algebra arguments.