Math and science::Algebra::Aluffi

Subgroups. Propositions.

The proofs of the following are on the reverse side. Can you remember them?

Let $φ : G \to H$ be a group homomorphism, and let $G^{'}$ be a subgroup of $G$ . Then the image of $G^{'}$ through $φ$ is a subgroup of $H$ .

Let $φ : G \to H$ be a group homomorphism, and let $H^{'}$ be a subgroup of $H$ . Then [what can be said about $φ^{- 1} (H)$ ?].

The proof has two steps, one for each of the requisite conditions of a subgroup.

Let $H_{1}$ and $H_{2}$ be two subgroups of $G$ . Then $H_{1} \cap H_{2}$ is a subgroup of $G$ .

24.03.2022