Some results of $\alpha$-coset groups
نویسندگان
1 Department of Mathematics, Khayyam University, Mashhad, Iran.
2 Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
3 Department of Mathematics, Khayyam University, Mashhad, Iran\ Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O.Box 1159, Mashhad, 91775, Iran.
doi
10.22034/as.2022.2806چکیده
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 2006.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 $p$.
