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

Abstract

AbstractAn element x of $$G^{*}$$ G ∗ 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 ∈ 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 ∈ 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 $$p^{j+1}$$ 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.