Let G1 and G2 be groups. Show that G1 x G2 is isomorphic to G2 x G1. Confused, p

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For simplicity, I will call G1 = G and G2 = H. PROOF. Let t: G x H --> H x G be a mapping such that t(g, h) = (h, g). The kernel of the map consists only of one element -- the identity element -- that is Ker(t) = {(g_e, h_e)} (since the map is simply a juxtaposition). Moreover, the image of the map is the entire set; that is, Im(t) = H x G (again, since the map is simply a juxtaposition). It follows from the First Isomorphism Theorem that... G x H ~ (G x H) / Ker(t) ~ Im(t) ~ H x G. []

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