On $m^{th}$-autocommutator subgroup of finite abelian groups

Subject Areas : History and biography

A. Gholamian ¹ , M. M. Nasrabadi ²

1 - Department of Mathematics, Birjand Education, Birjand, Iran | Farhangian University, Shahid Bahonar Campus, Birjand, Iran
2 - Department of Mathematics, University of Birjand, Birjand, Iran

Received: 2016-05-11 Accepted : 2016-08-24 Published : 2016-09-01

Keywords: ‎Automorphism‎, ‎Lower autocentral series‎, ‎Finite Abelian group,

Abstract :

Let $G$ be a group and $Aut(G)$ be the group of automorphisms of‎‎$G$‎. ‎For any natural‎number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $G$ is defined‎‎as‎: ‎$$K_{m} (G)=\langle[g,\alpha_{1},\ldots,\alpha_{m}] |g\in G‎,\‎alpha_{1},ldots,\alpha_{m}\in Aut(G)\rangle.$$‎‎In this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎‎all finite abelian groups‎.

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