On finite A-perfect abelian groups

Document Type : Research Paper


1 Department of Maths,birjand university

2 Department of mathematics, Birjand university, Birjand


‎Let $G$ be a group and $A=Aut(G)$ be the group of automorphisms of‎ ‎$G$‎. ‎Then the element $[g,\alpha]=g^{-1}\alpha(g)$ is an‎ ‎autocommutator of $g\in G$ and $\alpha\in A$‎. ‎Also‎, ‎the‎ ‎autocommutator subgroup of G is defined to be‎ ‎$K(G)=\langle[g,\alpha]|g\in G‎, ‎\alpha\in A\rangle$‎, ‎which is a‎ ‎characteristic subgroup of $G$ containing the derived subgroup‎ ‎$G'$ of $G$‎. ‎A group is defined as A-perfect‎, ‎if it equals its own‎ ‎autocommutator subgroup‎. ‎The present research is aimed at‎ ‎classifying finite abelian groups which are A-perfect‎.


Main Subjects

C. Chi$\check{s}$, M. Chi$\check{s}$, and G. Silberberg (2008). Abelian groups as autocommutator groups. Arch. Math. (Basel).. 90 (6), 490-492 P. Hegarty (1994). The absolute centre of a group. J. Algebra.. 169 (3), 929-935