Pullback and Pushforward Representations

In this section we show how to use group homomorphisms to construct a representation of one group from a representation of another group. 

First we show that (G, V, p o F) is a representation by checking the criteria given in Definition 4.7. We know by hypothesis that G is a group and V is a vector space. Because both p and 4 are group homomorphisms, it follows from Proposition 4.3 that p o F is a group homomorphism from G to QC (V). Hence p o 4 is a representation. 

Consider for example the inclusion map i of a subgroup G of a group G, defined in Exercise 4.37. By that exercise, the inclusion map is a group homomorphism. Note that for any representation p of G, the pullback representation p o z is just the restriction of p to the subgroup G. 

First we prove that p is well defined. Fix any g e G. Because is surjective, there is at least one g in the set we can use to define p(g). 

It remains to show that the value of p(g) does not depend on the choice of g e p- (g). Suppose gi, g2 e G and Kgi) = ( 2) = g- Wc must show that pCgi) = p(g2)- Note that 

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