Group Homomorphisms

**Bernhard** · September 6th 2011, 03:20 AM

I am trying to understand homomorphisms from $Z_n$ to $Z_k$ and despite some help from earlier posts am still struggling.

I am thus reading Beachy and Blair: Abstract Algebra on group homomorphisms and am trying to follow Example 3.7.7 on page 156 which seeks to determine all homomorphisms from $Z_n$ to $Z_k$ (see relevant page attached). I can follow this example but am having trouble with a particular step.

I will present their argument down to the problematic step - and I will make the example more concrete by seeking to determine all homomorphisms $\phi$ : $Z_6$ $\rightarrow$ $Z_{10}$

In terms of my particular example the argument of Beachy and Blair is as follows:

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Consider $\phi$ : $Z_6$ $\rightarrow$ $Z_{10}$

Any such homomorphism is completely determined by $\phi$ ( $[1]_6$ ), and this must be an element $[m]_{10}$ of $Z_{10}$ whose order is a divisor of 6.

In an abelian group with the operation denoted additively we have that if o(a) | n if and only if n.a = 0.

Applying this result to $[m]_{10}$ in $Z_{10}$ we have o( $[m]_{10}$ ) | 6 if and only if n. $[m]_{10}$ = $[0]_{10}$ which happens if and only if 10 | 6m (Beachy and Blair ask us to compare with Exercise 11 of Section 2.1 - see other attached sheet)

etc etc - see attached sheet
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My question is how does the step that concludes " which happens if and only if 10 | 6m" follow ie how does this last step of the argument follow?

I would be grateful for any help!

Peter

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