The Structure of p-fixed Autonilpotent Finite Groups

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 GK_n for a given group G, as GK_n=〈 [g ,α^n ] ┤| g∈ G,α∈Aut(G) 〉 which is characteristic subgroup of G . In this section, based on the induction method, we obtain the structure of GK_n 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 〈 (∏_(i=0)^(n-1)▒〖[g,α] [g,α,α]^i) [g,α,α,α]^((n(n-1)(n-2))/6) 〗┤|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 GK_n=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 Z_2 iff G is 1-fixed group also we show that abelian group G is p -fixed autonilpotent group iff it is isomorphic to G≅ C_(2^k ) where 1≤ k≤ 3