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\textbf{Homomorphisms}.
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Definition.  If $(G_1, *) $ and $(G_2, \diamond) $ are sets with operations $*$ and $\diamond$ respectively and $F:G_1 \rightarrow G_2$ is a function, then $F$ is called a \textit{homomorphism} if and only if
\begin{eqnarray*}
F(x * y) & = & F(x) \diamond F(y) .
\end{eqnarray*}

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Definition.  A homomorphism that is one-to-one is called an \textit{isomorphism}.

Exercise 7.1.  Prove that the following functions are homomorphisms (Caution: I think one of them is not!); which, if any, are isomorphisms:

\qquad a.  $f: (\mathbb{Z}, +) \rightarrow (\mathbb{Z}, +)$ where $f(x) = 5x$.

\qquad b.  $f: (\mathbb{R}, +) \rightarrow (\mathbb{R}, +)$ where $f(x) = 5x$.

\qquad c.  $f: (\mathbb{R}, \cdot) \rightarrow (\mathbb{R}, \cdot )$ where $f(x) = 5x$.

\qquad d.  $f: (\mathbb{R}, +) \rightarrow (\mathbb{R}, \cdot)$ where $f(x) = 2^x$.

\qquad e.  $f: (\mathbb{R}^+, \cdot) \rightarrow (\mathbb{R}, +)$ where $f(x) = \ln (x)$.

\qquad f.  $f: (\mathbb{C}, +) \rightarrow (\mathbb{R}, +)$ where $f(x+yi) = 3x + 5y$ ($\mathbb{C}$ denotes the complex numbers).

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Exercise 7.2.  Determine if the following are homomorphisms (isomorphisms), in each case make sure to check to see if the function is well-defined:

\qquad a.  $F: (\mathbb{Z}_5, +_5) \rightarrow (\mathbb{Z}_5, +_5)$ where $F([x]_5) = [2x+1]_5$.

\qquad b.  $F: (\mathbb{Z}_5, +_5) \rightarrow (\mathbb{Z}_5, +_5)$ where $F([x]_5) = [2x]_5$.

\qquad c.  $F: (\mathbb{Z}_{10}, +_{10}) \rightarrow (\mathbb{Z}_5, +_5)$ where $F([x]_5) = [2x]_5$.

\qquad d.  $F: (\mathbb{Z}_5, +_{5}) \rightarrow (\mathbb{Z}_{10}, +_{10})$ where $F([x]_5) = [2x]_5$.

\qquad e.  $F: (\mathbb{Z}_{31}, +_{31}) \rightarrow (\mathbb{Z}_{31}, +_{31})$ where $F([x]_{31}) = [7x]_{31}$.

\qquad f.  $F: (\mathbb{Z}_5, \cdot_5) \rightarrow (\mathbb{Z}_5, \cdot_5)$ where $F([x]_5) = [2x+1]_5$.

\qquad g.  $F: (\mathbb{Z}_5, \cdot_5) \rightarrow (\mathbb{Z}_5, \cdot_5)$ where $F([x]_5) = [2x]_5$.

\qquad h.  $F: (\mathbb{Z}_4, +_4) \rightarrow (\mathbb{Z}_5, \cdot_5)$ where $F([x]_4) = [2]_5^x$.

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Exercise 7.3.  Suppose $F: (\mathbb{Z}_5, +_5) \rightarrow (\mathbb{Z}_n, +_n)$ where $F([x]_5) = [ax+b]_n$ is a homomorphism.
Then:

1.) Either $a$ or $n$ is divisible by $5$.

2.) $b \sim_n 0$.

3.) $F({[0]}_5) = {[0]}_n$.

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Theorem 7.1  Suppose $F: (\mathbb{Z}_m, +_m) \rightarrow (\mathbb{Z}_n, +_n)$ is a homomorphism.  Then  $F({[0]}_m) = {[0]}_n$.

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%Exercise 7.4.  Let $S_n$ denote the permutation group on the elements  $\{1, 2, 3, \ldots, n\}$.  Determine if the following are homomorphisms:  in each case make sure to check to see if the function is well-defined:

%\qquad a.  $F: (\mathbb{Z}_5, +_5) \rightarrow (S_5, \cdot)$ where $F([x]_5) = (12345)^x$,

%\qquad b.  $F: (\mathbb{Z}_6, +_6) \rightarrow (S_6, \cdot)$ where $F([x]_6) = (123456)^x$,

%\qquad c.  $F: (\mathbb{Z}_5, +_5) \rightarrow (S_6, \cdot)$ where $F([x]_5) = (123456)^x$,

%\qquad d.  $F: (\mathbb{Z}_6, +_6) \rightarrow (S_5, \cdot)$ where $F([x]_6) = ((12)(345))^x$,

%\qquad e.  $F: (\mathbb{Z}_5 -\{0\}, \cdot_5) \rightarrow (S_4, \cdot)$ where $F([x]_5) = (1234)^x$,

%\qquad f.  $F: (\mathbb{Z}_8, +_8) \rightarrow (S_4, \cdot)$ where $F([x]_8) = (1234)^x$.


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