Math and science::Algebra::Aluffi

# Subgroups. Propositions.

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

### Image of subgroup is a subgroup in $$\cat{Grp}$$

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

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

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 $$G$$.