[[Representation theory MOC]]
# Square sum of irrep dimensions

As an immediate consequence of [[Orthonormality of irreps|orthogonality of irreps]],
it follows that the square sum of the dimensions of all (non-equivalent) irreps of a finite group $G$ equals the order $\abs G$ of the group. #m/thm/rep
$$
\begin{align*}
\sum_{\gamma \in \hat{G}} (d_{\gamma})^2 = \abs{G}
\end{align*}
$$

## Corollary

Since [[Irreps of abelian groups are 1-dimensional]],
it follows that the number of irreps $\abs{\hat{G}}$ of an abelian group $G$ equals the order of the group $\abs G$. #m/thm/rep 
$$
\begin{align*}
Z(G) = G \implies \abs{\hat{G}}= \abs G 
\end{align*}
$$


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