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Direct product of groups: Let (G, ?G) and (H, ?H) be groups, with identity elements eG...

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Advance Math

Direct product of groups: Let (G, ?G) and (H,?H) be groups, with identity elements eG andeH, respectively. Let g be any element of G, and h anyelement of H. (a) Show that the set G × H has a natural groupstructure under the operation (?G, ?H). Whatis the identity element of G × H with this structure? What is theinverse of the element (g, h) ? G × H? (b) Show that the mapiG : G ? G × H given by iG(g) = (g,eH) is a group homomorphism. Is it injective?Surjective? Do the same for the map iH : H ? G × H givenby iH(h) = (eG, h). (c) Show that the map?G : G × H ? G given by ?G (g, h) = g is agroup homomorphism. Is it injective? Surjective? Do the same forthe map ?H : G × H ? H given by ?H (g, h) =h. (d) Prove that the image of iG is the kernel of?H, and that the image of iH is the kernel of?G

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