Let $G$ be a finite group (not necessarily abelian), and consider the dual group $\hat{G}$, whose underlying set is the irreducible representations of $G$. My question is: What is the group structure on $\hat{G}$?
(I originally thought that by extending the underlying set to ``all representations'', I have a natural semi-ring structure on this set, namely the semi-ring of characters. However, this is not the notion of duality we want because the dual of the dual does not give us back anything that resembles the group)
(Another thought is that perhaps one can somehow exploit the group structure on $\mathbb{C}[G]$?)
Thank you for your help,
Maithreya
$\textbf{EDIT due to comments}$: I found where i read about this!: http://www.math.ucla.edu/~tao/247b.1.07w/notes10.pdf (I had lost it before) - See the end of page 5, and you see that the underlying set is indeed the collection of irreducible representations and he calls it the dual group - he says he will "get back to the group structure'' but I do not think he does.