Math 360, Fall 2019, Assignment 15

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

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.
4. Theorem relating kernels to injectivity.
5. Theorem concerning the normality of kernels.
6. Formula for the kernel of the canonical projection $\pi:G\rightarrow G/H$.
7. Theorem characterizing which subgroups can be the kernels of homomorphisms.

Solve the following problems:

1. Section 13, problems 1, 3, 5, 9, 28, 29, 44, 45, and 49.

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