On conjygacy classes in groups

Abstract

Let $G$ be a group. Write $G^{\ast }=G\setminus \{1\}$. An element $x$ of $G^{\ast }$ will be called deficient if $⟨x⟩<C_G(x)$ and it will be called non-deficient if $⟨x⟩=C_G(x).$ If $x\in G$ is deficient (non-deficient), then the conjugacy class $x^G$ of $x$ in $G$ will be also called deficient (non-deficient). Let $j$ be a non-negative integer. We shall say that the group $G$ has defect $j$, denoted by $G\in D(j)$ or by the phrase "$G$ is a $D(j)$-group", if exactly $j$ non-trivial conjugacy classes of $G$ are deficient. We first determine all finite $D(0)$-groups and $D(1)$-groups. Then we deal with arbitrary $D(0)$-groups and $D(1)$-groups: we find properties of arbitrary $D(0)$-groups and $D(1)$-groups, which force these groups to be finite.