(Solved): Clear and readable proofs please.  2. Let \( G \) be a group. Recall the direct product grou ...

Clear and readable proofs please. 

2. Let \( G \) be a group. Recall the direct product group \( G \times G \). Define an action \( (G \times G) C G \) given by: \[ \left(g_{1}, g_{2}\right) \cdot x=g_{1} x g_{2}^{-1}, \] for all \( \left(g_{1}, g_{2}\right) \in G \times G \) and \( x \in G \). (a) Prove that this action is valid, i.e., that it satisfies the axiom of a left action of \( G \times G \). (b) Describe the stabilizer \( (G \times G)_{e} \) of the identity element \( e \in G \), and give a proof of this fact. (c) Use the orbit-stabilizer theorem to compute \( \left|\mathcal{O}_{e}\right| \).