On D(j)-groups with an element of order \(p^{j+1}\) for some prime p

An element x of \(G^{*}\) $G_{}^{\ast }$ will be called deficient if \( \langle x\rangle < C_G(x)\) $⟨x⟩<C_G(x)$ and it will be called non-deficient if \(\langle x\rangle = C_G(x)\) $⟨x⟩=C_G(x)$ . If \(x\in G\) $x\in G$ is deficient (non-deficient), then the conjugacy class \(x^G\) $x^G$ 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 \(G\in D(j)\) $G\in D(j)$ or by the phrase “G is a D(j)-group", if exactly j non-trivial conjugacy classes of G are deficient. This paper deals with groups G which belong to D(j) for some positive integer j and which contain an element x of order \(p^{j+1}\) $p^{j+1}$ for some prime p. We determine all finite D(j)-groups. Then we prove that if such groups are locally graded, then they have to be finite.