I Understanding Proof: Clarifying the Relationship Between K and H

[Mr Davis 97]

https://imgur.com/a/jThCPLA

I'm trying to understand the proof here, and there is just one point that I get tripped up on. In the last paragraph, I'm not seeing exactly why $K\cap H < H$ based upon our choice of $y$. Could someone explain?

 

[fresh_42]

https://imgur.com/a/jThCPLA

I'm trying to understand the proof here, and there is just one point that I get tripped up on. In the last paragraph, I'm not seeing exactly why $K\cap H < H$ based upon our choice of $y$. Could someone explain?

The implicite (and missing) word is "proper". $H\cap K$ is always a subgroup of either. The fact that $y \notin H$ makes it a proper inclusion. $K\cap H \lneq H$ would have been the better sign.

 

[Mr Davis 97]

The implicite (and missing) word is "proper". $H\cap K$ is always a subgroup of either. The fact that $y \notin H$ makes it a proper inclusion. $K\cap H \lneq H$ would have been the better sign.

Oh, I think I got it now. One more question. Could this problem also be solved with the Fundamental Theorem of Finitely Generated Abelian Groups?

 

[fresh_42]

Help me, what theorem do you mean? The proof you have is quite short and straightforward, I don't know a better one.