Abstract
Let m be a positive integer and A an elementary abelian group of order q r with r ⩾ 2 acting on a finite q ′ -group G. We show that if for some integer d such that 2 d ⩽ r − 1 the dth derived group of C G ( a ) has exponent dividing m for any a ∈ A # , then G ( d ) has { m , q , r } -bounded exponent and if γ r − 1 ( C G ( a ) ) has exponent dividing m for any a ∈ A # , then γ r − 1 ( G ) has { m , q , r } -bounded exponent.
Highlights
Let A be a finite group acting coprimely on a finite group G
Any nilpotent group admitting a fixed-point-free automorphism of prime order q has nilpotency class bounded by some function h(q) depending on q alone
The result is a consequence of the classification of finite simple groups [21]: If A is a group of automorphisms of G whose order is coprime to that of G and CG(A) is nilpotent or has odd order, G is soluble
Summary
Let A be a finite group acting coprimely on a finite group G. Zelmanov’s techniques that led to the solution of the restricted Burnside problem [24] are combined with the Lubotzky– Mann theory of powerful p-groups [13], Lazard’s criterion for a prop group to be p-adic analytic [11], and a theorem of Bakhturin and Zaicev on Lie algebras admitting a group of automorphisms whose fixed-point subalgebra is PI [1] Another quantitative result of similar nature was proved in the paper of Guralnick and the second author [4]. Using the classification of finite simple groups it is shown in Section 4 that the A-invariant Sylow p-subgroups of G(d) are generated by their intersections with A-special subgroups of degree d This enables us to reduce the proof of Conjecture 1.3 to the case where G is a p-group, which can be treated via Lie methods. Throughout the article we use the term “{a, b, c, . . . }-bounded” to mean “bounded from above by some function depending only on the parameters a, b, c, . . . ”
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