## Some properties of a commutative diagram with exact rows

Consider the following commutative diagram of groups.

A commutative diagram of groups

Suppose the rows are exact; that is, for . Prove the following.

- If and are surjective and is injective, then is injective.
- If , , and are injective, then is injective.
- If , , and are surjective, then is surjective.
- If is surjective and and are injective, then is surjective.

- Let . Since is surjective, there exists such that . Now , so that , and so . Thus , and so . Say where . Since is surjective, there exists such that . Now , so that , and so . Since is injective, . In particular, , so that . Then . Thus , and so is injective.
- Let . Now , so that , so that . Thus . Since is injective, , so that . So . Thus there exists such that . Now , so that . Since and are injective, is injective, so that . Thus , and thus . Thus is injective.
- Let . Now . Since and are surjective, is surjective, so that there exists such that . Note that , so that . Thus , and so there exists such that . Since is surjective, there exists such that . Now . Thus , and so is surjective.
- Let . Now . Since is surjective, there exists such that . Now . Since is injective, . So , and so . Say such that . Now . Since is injective, . Thus is surjective.

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