Let m be a positive integer and A an elementary abelian group of order qr with r⩾2 acting on a finite q′-group G. We show that if for some integer d such that d2⩽r−1 the dth derived group of CG(a) has exponent dividing m for any a∈A#, then G(d) has {m,q,r}-bounded exponent and if γr−1(CG(a)) has exponent dividing m for any a∈A#, then γr−1(G) has {m,q,r}-bounded exponent.
Fixed points of coprime operator groups / Acciarri, C; Shumyatsky, P. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 342:01(2011), pp. 161-174. [10.1016/j.jalgebra.2011.06.013]
Fixed points of coprime operator groups
ACCIARRI C;
2011
Abstract
Let m be a positive integer and A an elementary abelian group of order qr with r⩾2 acting on a finite q′-group G. We show that if for some integer d such that d2⩽r−1 the dth derived group of CG(a) has exponent dividing m for any a∈A#, then G(d) has {m,q,r}-bounded exponent and if γr−1(CG(a)) has exponent dividing m for any a∈A#, then γr−1(G) has {m,q,r}-bounded exponent.
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