Question

Question: Let G and H be finite abelian groups. Prove that if$\mathit{G}\mathbf{\cong }\mathit{H}$ and only if G and H have the same invariant factors.

Short answer

It is proved that $G\cong H$if and only if$G$ and$H$ have same invariant factors.

Step-by-step solution

Theorem 9.12

Let G and H be finite abelian groups. Then, G is isomorphic to H if and only if G and H have the same elementary divisor.

Let $G$and$H$ be finite abelian groups.

Prove G≅H

Suppose$G$ and$H$ have the same invariant factors.

Therefore, we can say that$G$ and$H$ have the same elementary divisor.

Hence, byTheorem 9.12$G\cong H$ .

Prove G and H have same invariant factors

Suppose$G\cong H$ .

Now, byTheorem 9.12, we can say that $G$and $H$have same elementary divisor.

Therefore, we can also say that$G$ and$H$ have the same invariant factors.