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

An element x of G* G ∗ will be called deficient if x < CG(x) ⟨ x ⟩ < C G ( x ) and it will be called non-deficient if x = CG(x) ⟨ x ⟩ = C G ( x ) . If x∈ G x ∈ G is deficient (non-deficient), then the conjugacy class xG 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) G ∈ 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 pʲ⁺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.