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Recall the group Hom(G, A) (especially the group Hom(G, C ? ) whose elements are called...

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Advance Math

Recall the group Hom(G, A) (especially the group Hom(G, C ? )whose elements are called characters of G) and the group µn of n-throots of unity.

(i) Let n be a positive integer, prove that Hom(Cn, C ? ) ?= µn.Hints: let g be a generator of Cn. For every homomorphism ? : Cn ?C ? , prove that ?(g) ? µn (i.e. ?(g) is an n-th root of unity).Hence we have the map Hom(Cn, C ? ) ? µn given by ? 7? ?(g). Provethat this map is a group homomorphism and it is bijective.

(ii) Let G1 and G2 be groups, prove that Hom(G1 × G2, C ? ) ?=Hom(G1, C ? ) × Hom(G2, C ? ).

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