Browsing by Subject "Coprime action"
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Item Open Access On the influence of the fixed points of an automorphism to the structure of a group(Academic Press, 2021-01-06) Kızmaz, Muhammet YasirLet α be a coprime automorphism of a group G of prime order and let P be an α-invariant Sylow p-subgroup of G. Assume that p∉π(CG(α)). Firstly, we prove that G is p-nilpotent if and only if CNG(P)(α) centralizes P. In the case that G is Sz(2r) and PSL(2,2r)-free where r=|α|, we show that G is p-closed if and only if CG(α) normalizes P. As a consequence of these two results, we obtain that G≅P×H for a group H if and only if CG(α) centralizes P. We also prove a generalization of the Frobenius p-nilpotency theorem for groups admitting a group of automorphisms of coprime order.Item Open Access On the rank and exponent of the fixed points of coprime actions(Springer, 2022-02) Kızmaz, Muhammet YasirLet A be a group acting on a p-group P coprimely. We show that if A centralizes some specified abelian subgroups of P, then A acts trivially on P. As a consequence of this, we obtain that the special rank of CP(A) is strictly less than that of P unless the action of A on P is trivial. Secondly, we prove that if A acts on a group G coprimely and [G,A]=G, then the exponent of CG(A)/(CG(A))′ divides |G:CG(A)|.Item Open Access A short note on Isaacs–Navarro’s theorem(Birkhauser Verlag AG, 2019) Kızmaz, M. YasirIn this short note, we give a character free proof to a result of Isaacs–Navarro.