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Math Help - Homomorphism 4

  1. #1
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    Homomorphism 4

    If G1, G2 are two groups and G = G1 x G2 = {(a,b)|a in G1, b in G2, where we define (a,b)(c,d)=(ac,bd), show that:
    a) N={a,e2)|a in G1}, where e2 is the unit element of G2, is a normal subgroup of G.
    b) there is an isomorphism from N onto G1.
    c) there is an isomorphism from G/N onto G2.

    Please show steps. Thank you!
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  2. #2
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    Quote Originally Posted by mpryal View Post
    If G1, G2 are two groups and G = G1 x G2 = {(a,b)|a in G1, b in G2, where we define (a,b)(c,d)=(ac,bd), show that:
    a) N={a,e2)|a in G1}, where e2 is the unit element of G2, is a normal subgroup of G.
    b) there is an isomorphism from N onto G1.
    c) there is an isomorphism from G/N onto G2.

    Please show steps. Thank you!
    For (a) show gNg^{-1} = N for all g\in G.
    For (b) define f: N\to G_1 by f(a,e_2) = a.
    For (c) define f: G\to G_2 by f(a,b) = b now use fundamental homomorphism theorem.
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