1.6.9 Fields, Groups, Rings

Problem: Let \( (M, \cdot) \) - be a group, and \( \alpha \) and \( \beta \) - mappings from \( M \) to \( M \). Let's define an operation * on \( M \) by the rule: \( x * y=\alpha(x) \beta(y) \forall x, y \in M \). Prove that the conditions are equivalent: (1) \( (M, *) \) - a group; (2) \( \exists a, b \in M: \alpha(g)=g a, \beta(g)=b g \quad \forall g \in M \).