Math 361, Spring 2021, Assignment 1

Read:

1. Section 13.

Carefully define the following terms, then give one example and one non-example of each:

1. Homomorphism.
2. Monomorphism.
3. Epimorphism.
4. Pushforward (of a subgroup under a homomorphism; also known as the forward image of the subgroup).
5. Pullback (of a subgroup under a homomorphism; also known as the pre-image of the subgroup).
6. Image (of a homomorphism).
7. Kernel (of a homomorphism).
8. Canonical projection (from a group $G$ to a quotient $G/H$).

Carefully state the following theorems (you do not need to prove them):

1. Theorem characterizing pushforwards ("The pushforward of a subgroup is a...").
2. Theorem concerning pullbacks ("The pullback of a subgroup is a...").
3. Theorem relating images to surjectivity.

Solve the following problems:

1. Section 13, problems 1, 3, 5, 9, 28, 29, 45, and 49.
2. Prove the theorem concerning pullbacks.
3. Suppose that $G$ and $K$ are groups and $\phi:G\rightarrow K$ is a homomorphism. Prove that for any $H\leq G$, we have $H\subseteq \phi^{-1}[\phi[H]]$.
4. Give an example to show that strict containment may occur in the result above. (Hint: start with a homomorphism which is not injective.)
5. Suppose that $G$ and $K$ are groups and $\phi:G\rightarrow K$ is a homomorphism. Prove that for any $L\leq K$, we have $\phi[\phi^{-1}[L]]\subseteq L$.
6. Give an example to show that strict containment may occur in the result above. (Hint: start with a homomorphism which is not surjective.)

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