Shima Hosseini 

Department of Mathematics, Birjand Branch, Islamic Azad University, Birjand, Iran

DOI: https://doi.org/10.19184/jid.v26i1.42866

ABSTRACT 

The concept of autonilpotent groups was introduced by Moghaddam and Parvaneh in 2010 which is related to concepts of absolute centre and autocommutator subgroups that were first proposed by Hegarty in 1994 and 1997 respectively. In the present article we give some useful properties of such groups. In the first section, we define a subgroup GK𝑛 for a given group G, as GK𝑛=〈 [g ,αn] | g∈ G,α∈Aut(G) 〉 which is characteristic subgroup of 𝐺 . In this section, based on the induction method, we obtain the structure of GK𝑛 as 〈 [ g ,α ]𝑛 |g∈ G,α∈ Aut(G) 〉, 〈 [ g ,α ]𝑛 [g,α,α]n(n−1)/2|g∈ G,α∈ Aut(G) 〉 and 〈 (Π[g,α][g,α,α]𝑖) [g,α,α,α]n(n−1)(n−2)6𝑛−1𝑖=0|g∈ G,α∈ Aut(G) 〉 in autonilpotent groups of classes 2, 3 and 4 respectively. In second section, 𝑛-fixed group is introduced as a group G with GK𝑛=1 for some n∈ℕ. Then among other results of n-fixed groups, we characterize some n-fixed groups. Based on the role of the absolute centre subgroup in the structure of the group, we prove a non-trivial finite group G is isomorphic to ℤ2 iff G is 1-fixed group also we show that abelian group G is 𝑝 -fixed autonilpotent group iff it is isomorphic to G≅ C2𝑘 where 1≤ k≤ 3.

Keywords: Autonilpotent group, n-fixed group, autocommutator subgroup, absolute centre.