Math 360, Fall 2015, Assignment 10

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

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

1. Homomorphism.
2. Monomorphism.
3. Epimorphism.
4. Image (a.k.a. pushforward) (of a subset $C\subseteq A$ under a map $f:A\rightarrow B$).
5. Pre-image (a.k.a. pullback) (of a subset $D\subseteq B$ under a map $f:A\rightarrow B$).
6. Fiber (of a map $f:A\rightarrow B$ over a point $b\in B$).
7. Image (of a homomorphism).
8. Kernel (of a homomorphism).

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

1. Theorem concerning the image of a subgroup under a homomorphism.
2. Theorem concerning the pre-image of a subgroup under a homomorphism.
3. Theorem relating monomorphisms to kernels.

Solve the following problems:

1. Section 13, problems 1, 3, 4, 5, 9, 10, 16, 17, 19, and 21.
2. Suppose $\phi:G\rightarrow H$ is a homomorphism, and that $K\subseteq H$ is a normal subgroup of $H$. Show that the pre-image $\phi^{-1}[K]$ is normal in $G$.
3. Show by example that the corresponding statement for forward images is false, i.e. find groups $G$ and $H$, a homomorphism $\phi:G\rightarrow H$, and a normal subgroup $K\subseteq G$ such that $\phi[K]$ is not normal in $H$. (Hint: the smallest example occurs when $G=S_2$ and $H=S_3$.)

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