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How many homomorphisms are there of Z into $Z_2$?
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To count the number of homomorphisms from the group $\mathbb{Z}$ (the integers under addition) to the group $\mathbb{Z}_2$ (the integers modulo 2 under addition), we need to consider the properties of homomorphisms.
A homomorphism $f: \mathbb{Z} \rightarrow \mathbb{Z}_2$ preserves the group operation, which means:
$f(a+b)=f(a)+f(b)$
Since $\mathbb{Z}$ is a cyclic group generated by 1 , the homomorphism is completely determined by the image of the generator. Let's denote the image of 1 as $f(1)$. There are two possibilities for $f(1)$, either 0 or 1 in $\mathbb{Z}_2$.

1. If $f(1)=0$ in $\mathbb{Z}_2$, then for every integer $a, f(a)=a \cdot f(1)=a \cdot 0=0$. This defines the trivial homomorphism that maps every element of $\mathbb{Z}$ to 0 in $\mathbb{Z}_2$.

2. If $f(1)=1$ in $\mathbb{Z}_2$, then for every integer $a, f(a)=a \cdot f(1)=a \cdot 1=a$ in $\mathbb{Z}_2$. This defines the homomorphism that maps every element of $\mathbb{Z}$ to its equivalence class in $\mathbb{Z}_2$

So, there are two homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}_2$.

$\text{Number of homomorphisms } f: \mathbb{Z} \rightarrow \mathbb{Z}_2 \\ f(1) = 0 \\ f(1) = 1$

Thus, there are 2 homomorphisms from $\mathbb{Z}$ into $\mathbb{Z}_2$.
by Diamond (89,043 points)

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