Let $G$ be a finite group with $|G|=n$. In the case $Z(G)=\lbrace e \rbrace$, the following equation seems to hold:
\begin{equation} n(n-1)=\sum_{\substack{\lambda_i \in \lambda(n-1)\\ \lambda_i>1}}\lambda_i^2+\sum_{a \in G \setminus \lbrace e \rbrace}(|C_G(a)|-1)+\sum_{a \in G \setminus \lbrace e \rbrace}|C_G(a)|^2([G:C_G(a)]-1) \end{equation}
where $C_G(a)$ is the centralizer of $a$ in $G$ and $\lambda(n-1)$ is a partition of the integer $n-1$.
Before anyone wastes time going through my proof's sketch, can some more trained eye than mine acknowledge at a glance whether above equation is:
- trivial (e.g. an identity);
- correct and well-known;
- clearly wrong.