Math 360, Fall 2017, Assignment 11

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Read:

1. Section 13.
2. Section 14.

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

1. Canonical projection (from $G$ to $G/H$).
2. Homomorphism.
3. Monomorphism.
4. Epimorphism.
5. Pushforward (or forward image or just image) of a subgroup under a homomorphism.
6. Pullback (or inverse image or pre-image) of a subgroup under a homomorphism.

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

1. Theorem concerning whether $G/H$ is a group.
2. Theorem characterizing canonical projection as a certain type of mapping.
3. Theorem concerning pushforwards and pullbacks of subgroups.

Solve the following problems:

1. Section 13, problems 1, 2, 3, 6, 8, 9, 10, 25, 26, and 27 (in problems 1-10, if the given map is a homomorphism, please also say whether it is a monomorphism and/or an epimorphism).
2. (Sign morphism) Recall the map $\mathrm{sgn}:S_n\rightarrow \mathbb{R}^*$ sending even permutations to $1$ and odd permutations to $-1$. Show that $\mathrm{sgn}$ is a homomorphism into the multiplicative group $(\mathbb{R}^*,\cdot)$. Is it a monomorphism? An epimorphism?
3. Consider the canonical projection map $\pi:\mathbb{Z}\rightarrow\mathbb{Z}_6$. Describe the pushforward $\pi[\left\langle 4\right\rangle]$. Then describe the pullback $\pi^{-1}[\left\langle 2\right\rangle]$. Do pushforward and pullback always undo each other?

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