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:

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

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

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.
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.
If $\phi$ is a homomorphism from $G$ to $H$, then $\phi(e_G)=e_H$.
If $\phi$ is a homomorphism from $G$ to $H$, then $\phi(a^{-1}=\phi(a)^{-1}$