# Induction commutes with duals and tensor products

I need references for the following two results. Let $G$ be a finite group and let $H$ be a subgroup of $G$. Let $V$ be a finite-dimensional representation of $H$.

1. $(\text{Ind}_H^G V)^{\ast} \cong \text{Ind}_H^G V^{\ast}$.

2. Given a finite-dimensional representation $W$ of $G$, we have $(\text{Ind}_H^G V) \otimes W \cong \text{Ind}_H^G (V \otimes \text{Res}_H^G W)$.

I know references for these that prove them with character theory, but I need them for general fields (not just fields of characteristic $0$).

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Is $G$ supposed to be finite? In any case, I would imagine that these kinds of results follow from Frobenius reciprocity. – Qiaochu Yuan Aug 8 '12 at 16:46
The group $G$ is finite. However, the version of Frobenius reciprocity I know (eg the one in Serre's book) is a theorem about characters, so I don't think it applies here. – Juan Aug 8 '12 at 16:51
Frobenius reciprocity lifts to the isomorphism $\text{Hom}_G(\text{Ind}_H^G V, W) \cong \text{Hom}_H(V, \text{Res}_H^G W)$, and this holds in all characteristics (it is a special case of the tensor-hom adjunction (en.wikipedia.org/wiki/Tensor-hom_adjunction) and can in fact be used to define the induced representation). – Qiaochu Yuan Aug 8 '12 at 17:23
@QiaochuYuan : Thanks. I just spent 15-20 minutes trying to prove these with that version of Frobenius reciprocity. I was unsuccessful, but that doesn't mean it can't be done. – Juan Aug 8 '12 at 18:38