Some results of \alpha-coset groups

We call G to be an \alpha-coset group, if it contains a proper \alpha-invariant normal subgroup N such that Nx^\alpha =\{x^g~|~ g\in G\}, for some automorphism \alpha of G and any x \in G\setminus N. Clearly, if \alpha is identity automorphism of G, one obtains the notion of con-cos groups, which was first introduced by Muktibodh in ۲۰۰۶.In the present article, we discuss some properties of the new notion. Also, we introduce the concept of \alpha-Camina groups and give its connection with the groups of property \mathcal P, where \mathcal P is the class of all finite groups such that their \alpha-centres are the same as \alpha-commutator subgroups of order p.

Keywords:

Absolute centre , alpha-centre , alpha-commutator subgroup , Central automorphism , Left and right alpha-commutator