Exponent of a finite group with a fixed-point-free four-group of automorphisms.
Let \(G\) be a finite group acted on by the four-group \(V\simeq C_2\times C_2\) in such a manner that \(C_G(V)=1\) (in particular \(G\) has odd order). An earlier theorem of the author [Monatsh. Math. 168, No. 1, 113-124 (2012; Zbl 1270.20026), Theorem 1.5] states that if the centralizers \(C_G(v_1)\) and \(C_G(v_2)\) of two distinct elements of \(V\) have exponent \(e\), then the exponent of \(G'\) is bounded in terms of \(e\) only. In this paper it is proved that if \(C_G(v)\) has exponent \(e\) for some \(v\in V\), then the exponent of \(G''\) is bounded in terms of \(e\) only.
scientific article; zbMATH DE number 6176745
scientific article