On groups with same number of centralizers

Abstract

In this paper, among other results, we give some sufficient conditions for every non-abelian subgroup of a group to be isoclinic with the group itself. It is also seen that under certain conditions, two groups have same number of element centralizers implies they are isoclinic. We prove that if $G$ is any group having $4, 5, 7$ or $9$ element centralizers and $H$ is any non-abelian subgroup of $G$, then $\mid \Cent(G)\mid=\mid \Cent(H)\mid$ and $ G' \cong H' \cong C_2, C_3, C_5$ or $C_7$ respectively. Furthermore, it is proved that if $G$ is any group having $n \in \lbrace 4, 5, 6, 7, 9 \rbrace$ element centralizers, then $\mid G' \mid=n-2$.