Abstract. The order of an element x in a finite group G is the smallest positive integer k, such that k x is the group identity. The set of all possible such orders joint with the number of elements that each order referred to, is called the order classes of G. The conjugacy classes of dihedral groups already known, the conjugacy classes is a refinement partition to the order classes. In this paper, the order classes of dihedral groups are derived. In addition, clarifications for some cases related to the size of the groups were given.

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Let M be a semiprime Γ-ring with involution satisfying the condition that aαbβc = aβbαc (a, b, c ∈ M and α, β ∈ Γ). An additive mapping. In this paper we will prove that if d is Γ * -derivation of a semiprime Γ-ring with involution which is either an endomorphism or anti-endomorphism, then d=0.

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