# I Understanding a proof

#### 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?

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#### fresh_42

Mentor
2018 Award
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

Mentor
2018 Award
Help me, what theorem do you mean? The proof you have is quite short and straightforward, I don't know a better one.

"Understanding a proof"

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