Exercise 976Algebra › GroupsL1Added 3 Nov 2026

Isomorphism criteria

#isomorphism#bijective homomorphism#criterion

Statement

Let $f:G\to G'$ be a homomorphism of finite groups with $|G|=|G'|$ . Show that $f$ is an isomorphism iff $f$ is injective iff $f$ is surjective.

Hint

Use $|G|=|\ker f|\cdot|\operatorname{Im} f|$ .

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Solution

Counting. For any homomorphism $f$ , by the first isomorphism theorem, $|G|=|\ker f|\cdot|\operatorname{Im} f|$ .

Injectivity. $f$ injective $\iff\ker f=\{e\}\iff|\ker f|=1\iff|\operatorname{Im} f|=|G|=|G'|$ . Since $\operatorname{Im} f\subseteq G'$ and $|G'|=|G|$ , we get $\operatorname{Im} f=G'$ .

Surjectivity. Similarly, $\operatorname{Im} f=G'\iff|\operatorname{Im} f|=|G'|=|G|\iff|\ker f|=1\iff$ injective.

Conclusion. For finite groups of the same order, injective $\Leftrightarrow$ surjective $\Leftrightarrow$ isomorphism.

Isomorphism criteria

Statement

Solution

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