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

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-th roots 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 α : Cn → C ∗ , prove that α(g) ∈ µn (i.e. α(g) is an n-th root of unity). Hence we have the map Hom(Cn, C ∗ ) → µn given by α 7→ α(g). Prove that 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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Colby Messinger answered 2 years ago

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