Math 360, Fall 2014, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Direct product (of two groups).
2. Direct sum (of two abelian groups written additively).
3. Homomorphism.
4. Monomorphism.
5. Epimorphism.

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

1. Theorem relating $\mathbb{Z}_m\times\mathbb{Z}_n$ to $\mathbb{Z}_{mn}$ (Theorem 11.5 in the text).
2. Fundamental Theorem of Finitely Generated Abelian Groups (Theorem 11.12).
3. Theorem concerning the value of a homomorphism at the identity.
4. Theorem concerning the value of a homomorphism on an inverse.

Solve the following problems:

1. Section 11, problems 1, 3, 8, 23, and 25.
2. Section 12, problems 1, 5, and 7.

--------------------End of assignment--------------------

Solutions:

Let $(G,\ast )$ and $(H, \ast \ast)$ be groups. Their direct product $G \times H$ is the set $G \times H$ under the operation $( g_1, h_1 )(g_2, h_2)=(g_1 \ast g_2 , h_1 \ast \ast h_2)$.