The Theorem states that:
$\left | G \right |=\left | Z(G) \right |+\sum_{g'}\frac{\left | G \right |}{\left | Z(g) \right |}$
The author goes on to state a Lemma: If $S(a)$ is the conjugacy class containing $a \in G$ and $\gamma(a) =$ number of elements in the conjugacy class $S(a)$ then $ \gamma(a)=\frac{\left | G \right |}{\left | Z(a) \right |}$
He further goes on to proving two claims to prove the lemma:
If $G= g_{1}Z(a)+...+g_{\gamma}Z(a), \gamma = \gamma(a)[G:Z(a)]$
Claim 1: Any conjugate $xax^{-1}$ of $a$ , $(x \in G)$ coincides with one and only one if $g_{i} a g_{i}^{-1}=a$
Claim 2 The $\gamma$ conjugates of $a$ are distinct from one another
Though I do understand the claim 2, but I can't understand the relevance or necessity of claim1. My idea is if we can prove that each element $x$ is indeed mapped and uniquely in one $g_iZ(a)$ then the job should be done. My question is what is the "behind the scenes" relevance for claim 1
Soham