Let G and H be two groups. The groups G x Hand H x G are isomorphica)for any G and any H.b)only if one of them is cyclic.c)only if one of them is abelian.d)only if G and H are isomorphic.Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation:

To prove the correct answer, we need to show that if G × H and H × G are isomorphic, then G and H are isomorphic.

Proof:

Let f: G × H → H × G be an isomorphism between the two groups.

Step 1: G is isomorphic to H
Consider the map π₁: G × H → G defined by π₁(g, h) = g. This map is a projection onto the first coordinate.

Similarly, consider the map π₂: H × G → G defined by π₂(h, g) = g. This map is a projection onto the second coordinate.

Since f is an isomorphism, it preserves the group structure. Therefore, the composition of f with the projections π₁ and π₂ gives:

f ◦ π₁: G × H → H × G → G
f ◦ π₂: H × G → G × H → H

Step 2: f ◦ π₁ and f ◦ π₂ are isomorphisms
We need to show that f ◦ π₁ and f ◦ π₂ are isomorphisms.

Step 3: f ◦ π₁ is an isomorphism
To prove that f ◦ π₁ is an isomorphism, we need to show that it is a bijection and preserves the group operation.

Injectivity: Suppose (g₁, h₁) and (g₂, h₂) are two elements in G × H such that f ◦ π₁(g₁, h₁) = f ◦ π₁(g₂, h₂). This implies that f(g₁) = f(g₂). Since f is an isomorphism, it is injective, so g₁ = g₂. Thus, (g₁, h₁) = (g₂, h₂), and f ◦ π₁ is injective.

Surjectivity: Let (h, g) be an element in H × G. Consider the element (g, h) in G × H. Since f is surjective, there exists an element (g, h) in G × H such that f(g, h) = (h, g). Therefore, f ◦ π₁ is surjective.

Preservation of the group operation: Let (g₁, h₁) and (g₂, h₂) be two elements in G × H. Then:

f ◦ π₁((g₁, h₁) · (g₂, h₂)) = f ◦ π₁(g₁g₂, h₁h₂) = f(g₁g₂) = f(g₁) · f(g₂) = f ◦ π₁(g₁, h₁) · f ◦ π₁(g₂, h₂)

Therefore, f ◦ π₁ preserves the group operation.

Step 4: f ◦ π₂ is an isomorphism
Using similar arguments as in Step 3, we can show that f ◦ π₂ is an isomorphism.

Step 5: G is isomorphic to H
Since f ◦ π₁ and f ◦ π₂ are isomorphisms,

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