TITLE: What is the mistake when I write down the class equation?
QUESTION [4 upvotes]: Let $(G,\cdot)$ be a finite group with $n$ elements which has a trivial center.
If we consider $G=\{e, x_1, x_2, ..., x_{n-1} \}$, then from the class equation we have that $$n=1+ \sum_{k=1}^{n-1} [G:C(x_k)].\tag{1}$$
By Lagrange's theorem we have that $n=|C(x_k)|\cdot [G:C(x_k)], \forall k=\overline{1,n-1}$, so $(1)$ rewrites as $$n=1+\sum_{k=1}^{n-1}\frac{n}{|C(x_k)|}.\tag{2}$$
If we let $M=\max\{|C(x_k)| | k=\overline{1,n-1}\}$(this is possible since these sets are finitee), we have that $\frac{n}{|C(x_k)|}\ge  \frac{n}{M}$, so $(2)$ implies that $$n-1\ge \frac{n(n-1)}{M}\iff M\ge n\tag{3}$$
and since $M\le n$, we obtain that $M=n$, which contradicts the fact that $Z(G)=\{e\}$.
Where did I go wrong?

REPLY [4 votes]: You don't have $n-1$ distinct conjugacy classes(why?). Thus, you cannot take the sum from $1$ to $n-1$ in Equation $(1)$.
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