Group Isomorphisms and Automorphisms Practice Questions

Group isomorphisms and automorphisms describe the structure and symmetry of groups.

• A group isomorphism is a one-to-one mapping between two groups that preserves the group operation, showing that the two groups have the same algebraic structure.
• An automorphism is an isomorphism of a group onto itself, representing the group's internal symmetry.

Question 1: Prove that (Z,+) and (2Z,+) are isomorphic.

• Define f:Z→2Z by f(x) = 2x.
• One-one: f(x) = f(y) ⇒ 2x=2y ⇒ x=y.
• Onto: For every y∈ 2Z, there exists x∈ Z such that y = 2x.
• Homomorphism: f(x+y) = 2(x+y) = 2x+2y = f(x)+f(y).
• Hence, f is an isomorphism.

Question 2: Show that (R⁺ , ⋅) and (R, +) are isomorphic via f(x) = log(x).

• One-one: f(x) = f(y) ⇒ log(x) = log(y) ⇒ x = y.
• Onto: For every y∈ R, there exists x ∈ R⁺ such that y = log(x).
• Homomorphism: f(x⋅y) = log(x⋅y) = log(x)+log(y) = f(x)+f(y).
• Hence, f is an isomorphism.

Question 3: Verify that the groups (Z₆, +₆) and (Z₂×Z₃, +) are isomorphic.

• Define f: Z₆→Z₂×Z₃ by f(x) = (xmod2,xmod3).
• One-one: Suppose f(x) = f(y). Then (xmod2, xmod3) = (ymod2,ymod3). This implies x ≡ ymod6, hence x = y.
• Onto: For every 𝑍₃(a,b) ∈ Z₂×Z₃, there exists x∈Z₆ such that xmod2 = a and xmod3 = b.
• Homomorphism: f(x+₆​y) = ((x+y)mod2,(x+y)mod3) = f(x)+f(y).

Hence, f is an isomorphism.

Question 4: Check if the function f(x) = x²defines an isomorphism between the groups (R,+) and (R⁺,⋅).

• Define f:R→R⁺ by f(x) = x².
• One-one: Suppose f(x) = f(y). Then x² = y².This implies x = ±y, so f is not one-one.
• Onto: For every y∈R⁺, choose x = √y ​or x = − √y in R.
• Homomorphism: f(x+y) = (x+y)² = x² + 2xy + y² ≠ f(x)⋅f(y).

Since f is neither one-one nor a homomorphism, it is not an isomorphism.

Practice Problems

1. Prove that the groups (Z₄,+₄) and (Z₂×Z₂,+) are isomorphic.

2. Show that (R⁺,⋅) and (R,+) are isomorphic via f(x) = e^x.

3. Verify that the groups (Z₆,+₆) and (Z₂×Z₃,+) are isomorphic.

4. Determine if the mapping f(x)=2x from (R,+) to (R,+) is an automorphism.

5. Check if the function f(x) = x² defines an isomorphism between the groups (R,+) and (R⁺,⋅).