Question

Let g1 and g2 be groups and let f: g1g2 be an isomorphism. prove that g1 is abelian, then g2 is abelian.

Answer 1

If g1 is abelian, an isomorphism f from g1 to g2 preserves the group structure, ensuring that g2 is also abelian.

Step-by-step explanation:

To show that if g1 is abelian, then g2 is abelian given an isomorphism f: g1 → g2, we rely on the properties of isomorphisms. An isomorphism is a bijective homomorphism, which means it preserves the group structure between two groups. For any elements a, b in g1, if g1 is abelian, a * b = b * a. Applying the isomorphism f, we have f(a * b) = f(b * a). Because f is a homomorphism, f(a * b) = f(a) * f(b) and f(b * a) = f(b) * f(a). Hence, f(a) * f(b) = f(b) * f(a) for all elements in g2, which implies that g2 is abelian.