Necessary and Sufficient Conditions Where One Γ-Semigroup is a Γ-Group

In this study, researchers have studied the Γ-algebraic structures and some characteristics of them. According to Sen and Saha, we defined algebraic structures: Γ-semigroup, Γ-regular semigroup, Γ-idempotent semigroup, Γ-invers semigroup and Γ-group. Theorem 2, 3 and 4 proves the existence of Γ-group and gives necessary and sufficient conditions where one Γ-semigroup is a Γ-group. Finally, theorem 5 shows necessary and sufficient conditions where one Γ-regular semigroup is a Γ-group. In addition, for every Γ- algebraic structure that we mentioned before we give an original example.