On 10-Centralizer Groups of Odd Order

Abstract

Let be a group, and let |Cent( )| denote the number of distinct centralizers of its elements. A group is called -centralizer if |Cent( )| = . In this paper, we investigate the structure of finite groups of odd order with |Cent( )| = 10 and prove that there is no finite nonabelian group of odd order with |Cent( )| = 10.

Groups with Exactly Ten Centralizers

“…This result also was proved independently (in odd order case) by Foruzanfar and Mostaghim in [8], they prove that, there is no finite nonabelian group of odd order with |Cent(G)| = 10.…”

Counting the Number of Centralizers of 2-Element Subsets in a Finite Group

“…1 and by Theorems 3.1(2) and 1.2(10), |2 − Cent(G)| − 1 = |Cent(G)| = n + 2. Now by Theorems 3.1(2) and 3.4, |2 − Cent(Z n ⋊ Z p )| − 1 = |Cent(Z n ⋊ Z p )| = n |Z(Zn⋊Zp)| + 2.This completes the proof.Corollary 3.6.…”

“…(8) G is primitive 9−centralizer if and only if G Z(G) ∼ = D 14 , Hol(Z 7 ) or a non-abelian group of order 21 [5]. (9) There is no 10−centralizer groups of odd order [10]. (10) If G is a primitive 11−centralizer group of odd order, then G Z(G) ∼ = (Z 9 × Z 3 ) ⋊ Z 3 [15].…”

“…(9) There is no 10−centralizer groups of odd order [10]. (10) If G is a primitive 11−centralizer group of odd order, then G Z(G) ∼ = (Z 9 × Z 3 ) ⋊ Z 3 [15].…”