Representation theory question + SU(n)

A- Let $H$ be a subgroup of a finite group $G$.Let $\alpha$ and $\beta$ be class function of $G$ and $H$ respectively. Prove that $$Ind^{G}_{H} (\beta . Res^{G}_{H})= \alpha .Ind^{G}_{H}(\beta)$$

B- Show that

1.Each element in $SU(n)$ conjugate to a diagonal matrix.

1. Every character of $SU(n)$ is uniquely determined by its restriction to subgroup $T$ consisting of all diagonal matrices whose diagonal coefficients have absolute value 1.

Many thanks.

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Could you explain the notation please? – joriki Apr 2 '11 at 19:07
Ind is the induced representation Res is the restricted representation – user8968 Apr 2 '11 at 19:15
I think it means the image of Ind:R(H)->R(G)is an ideal of the ring R(G)... Is that right or not? .. if yes, how can I show that?? Also if not , please help me to solve it :) – user8968 Apr 2 '11 at 19:21
Dear Martin, I think there is an $\alpha$ missing on the left side of your formula. Also, your two questions are completely unrelated (as far as I can tell), so I would suggest asking them separately. – Matt E Apr 3 '11 at 2:29