(RESOLVED) Algebra help: decomposition of finite abelian groups

[burnt sigil]

I have been trying this problem for a bit now, and it seems incredibly obvious, but I am not sure how to formalize it. All of the proofs I've seen online use log_p logarithms which seems overkill, plus we haven't covered that in class.
For reference, (1c) is showing that Hom(Z/n, Z/m) \cong Z/d.

[hazy falcon]

Just put the fries in the bag bro

[severe sparrow]

I think i know a nice proof for this for modules

i think you can make it work for groups give me a second

Start by thinking about a homomorphism from Z_n to Z_m with \phi(1)=a, and \phi(0)=\phi(n)=n\phi(1)=na=0

wait nvm 😭 that doesnt work still

i think any proofs of this for groups is going to be a bit involved, i dont think i can do it

[snow wyvern]

U need to put an application

Then show it that is a morphisme first

After u need to prove that ker(z/n,Z/m)={0}

So it's injectif

And for surjection is obv no need for proof

[burnt sigil]

two of y'all are dumb af 🙏

thanks to the one person who tried 🙏

i figured it out also so the question is resolved

(RESOLVED) Algebra help: decomposition of finite abelian groups

[brisk hound]

consider marking the post as .solved to do this

[burnt sigil]

.solved

[timber lava]

Solved

Post marked as solved by @burnt sigil.

Use .unsolved if this was a mistake.

[burnt sigil]

thanks