Desmond Russo

*I didn't answer. I only looked at him coldly and I put the knife in my jacket pocket. I took off my leather jacket and hung it on the coat rack and looked at him. My gaze was sharp, as if it had its own meaning. I stepped closer to Shoto's crib, holding him carefully so as not to wake him up.* Come back to Moscow with Shoto *I said trying to firm while handing over Shoto's small body wrapped in a warm blanket. I stared for a moment at Shoto's cute face who was still sleeping in Desmond's arm.*

To determine whether the function \( f: G \to G \) defined by \( f(x) = x^{-1} \) is a group homomorphism, we analyze its behavior under the group operation and its relation to commutativity. **Proof:** 1. **Forward Direction (\( \Rightarrow \)):** Assume \( f \) is a homomorphism. Then for all \( a, b \in G \), \[ f(ab) = f(a)f(b) \implies (ab)^{-1} = a^{-1}b^{-1}. \] By the property of inverses in a group, \((ab)^{-1} = b^{-1}a^{-1}\). Thus: \[ b^{-1}a^{-1} = a^{-1}b^{-1}. \] Taking the inverse of both sides: \[ (b^{-1}a^{-1})^{-1} = (a^{-1}b^{-1})^{-1} \implies ab = ba. \] Hence, \( G \) is abelian. 2. **Converse (\( \Leftarrow \)):** Assume \( G \) is abelian. For all \( a, b \in G \), \( ab = ba \). Then: \[ f(ab) = (ab)^{-1} = b^{-1}a^{-1} = a^{-1}b^{-1} \quad (\text{since } G \text{ is abelian}). \] Therefore, \( f(ab) = f(a)f(b) \), so \( f \) is a homomorphism. **Conclusion:** The function \( f(x) = x^{-1} \) is a group homomorphism if and only if \( G \) is abelian. \(\boxed{f \text{ is a homomorphism } \iff G \text{ is abelian}}\)

"I-"*Before I even spoke, one of your companions grab me, mistaking me for a prey..* Joon-Hoo:"Boss! I got the target! Let's go!"

*he turns to his companion, his expression darkens as he sees him holding you, his body tenses and his fists clench, he knows he can't let his companion take you, not after what happened with your friends, he looks back at you and his eyes soften again* "Let her go."

Joon-Hoo:"But boss! She's the target!"

*Desmond smiles back at you and waves goodbye. he can't help but feel a sense of happiness and contentment as he walks out the door. he's finally back with the woman he loves, and nothing else matters.*