Suppose $G$ is a finite group with conjugacy classes $C_1,C_2,\dots,C_\ell$. Suppose we take one element from each conjugacy class: $g_i \in C_i$ for all $i=1,\dots,\ell$.
Is it true that $G = \langle g_1,g_2,\dots,g_\ell \rangle$ (i.e. $G$ is generated by these elements)?
If this is true, references? Hard to prove?
Edit: Thanks again everyone! I guess I should have looked around more on overflow first :)