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:

Definitions:

1. Let $(G,\ast )$ and $(H, \vartriangle)$ be groups. Their direct product $G \times H$ is the set $G \times H$ under the operation $( g_1, h_1 )(g_2, h_2)=(g_1 \ast g_2 , h_1 \vartriangle h_2)$.
3. Let $(G, \ast)$ and $(H, \vartriangle)$ be groups. A homomorphism from $G$ to $H$ is a function $\phi : G \longrightarrow H$ that preserves $\phi ( g_1 \ast g_2) = \phi (g_1) \vartriangle \phi (g_2)$.
4. A monomorphism is a homomorphism that is injective.
5. An epimorphism is a homomorphism that is surjective.

Theorems:

1. The group $Z_m \times Z_n$ is cyclic and is isomorphic to $Z_{mn}$ if and only if $m$ and $n$ are relatively prime, that is, the gcd of $m$ and $n$ is 1.
2. Every finitely generated abelian group is isomorphic to a direct product of cyclic groups of the form \begin{equation*} \prod_{i=1}^{k}\mathbb{Z}_{p_i^{r_i}} \times \prod_{j=1}^n \mathbb{Z} \end{equation*} where the $p_i$ are primes (not necessarily unique) and the $r_i$ are positive integers. The factorization is unique except for a reordering of the factors, that is the $(p_i)^{r_i}$ are unique and the integer $n$ (the Betti number of $G$) is unique.
3. If $\phi$ is a homomorphism from $G$ to $H$, then $\phi(e_G)=e_H$.
4. If $\phi$ is a homomorphism from $G$ to $H$, then $\phi(a^{-1}=\phi(a)^{-1}$