On D(j)-groups with an element of order pj+1 for some prime p

An element x of G∗ will be called deficient if ⟨x⟩<C_G(x) and it will be called non-deficient if ⟨x⟩=C_G(x). If x∈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∈D(j) or by the phrase “G is a D(j)-group", if exactlyj non-trivial conjugacy classes of G are deficient. This paper deals with groups G which belong to D(j) for some positive integerj and which contain an element x of order 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.