Autocentralizer automorphisms of groups

Subject Areas : Group theory

F. Karimi ¹ , M. M. Nasrabadi ² , A. Kaheni ³

1 - Department of Mathematics‎, ‎Faculty of Mathematical Science and Statistics‎, ‎University of Birjand‎, ‎Birjand‎, ‎Iran
2 - Department of Mathematics‎, ‎Faculty of Mathematical Science and Statistics‎, ‎University of Birjand‎, ‎Birjand‎, ‎Iran
3 - Department of Mathematics‎, ‎Faculty of Mathematical Science and Statistics‎, ‎University of Birjand‎, ‎Birjand‎, ‎Iran

Received: 2024-07-30 Accepted : 2024-09-25 Published : 2024-09-26

Keywords: Centralizer subgroup, autocentralizer subgroup, autocentralizer automorphism,

Abstract :

‎Let $G$ be a group and $C_G(a)$ be a normal centralizer subgroup of $G$ for some $a\in G.$ Assume that‎, ‎$\Upsilon^a _2(G)= [G‎, ‎C_G(a)]$ and $Aut^{\Upsilon^a _2(G)}_{Z(C_G(a))}(G)$ is the set of all automorphisms of $G$ that centralizes $\frac {G}{\Upsilon ^a_2(G)}$ and $Z(C_G(a)).$ In this paper‎, ‎we focus on the group $Aut^{\Upsilon^a _2(G)}_{Z(C_G(a))}(G)$ and try to characterize its properties‎.

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