I am trying to understand homomorphisms from to 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 to (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 :
In terms of my particular example the argument of Beachy and Blair is as follows:
Any such homomorphism is completely determined by ( ), and this must be an element of 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 in we have o( ) | 6 if and only if n. = 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
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!