求助

koukou8617

高手能不能提供完整的步骤阿
1。Suppose G and H are groups. Consider the set G x H under the binary operation •defined by  (g_1, h_1) •(g_2,h_2) = (g_1g_2, h_1h_2)
where g_1, g_2 ∈G and h_1, h_2 ∈ H
(a) Prove that G x H is a group under the operation •
(b) Prove that G x H is isomorphic to H x G.
(c) Prove that G x{1_H} is isomorphic to G, where 1_H is the identity element of H.
(d) Prove that (G x{1_H})is normal subgroup of (G x H).
(e) Prove that (G x H)/( G x{1_H}) is isomorphic to H.

2。Given G a group, let G x G be the group under the operation • defined  (g_1, g_1) •(g_2,g_2) = (g_1g_2, g_1g_2)
Set Δ_G = {(g; g) : g ∈G} belongs to G x G.
(a) Prove that Δ_G is isomorphic to G
(b) Prove thatΔ_G is normal subgroup to G x G if and only if G is abelian.

3。Let C^x denote the set of all non-zero complex numbers under multiplication. (Recall that a complex number is of the form a + bi, where a and b are real numbers and i is defined by the relation i^2 = -1.) Define the map phi : C^x C^x  by phi(z) = z/|z|
(a) Prove that phi is a homomorphism.
(b) What is the image of phi? Describe it geometrically.
(c) What is the kernel of phi?
(d) Let R_>0 denote the group of positive real under multiplication, and let S^1 be the group{z∈C^x : |z |= 1}
under multiplication. Prove that C^x/R_>0 is isomorphic to S^1
.
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