I Understanding a proof

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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

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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.
 
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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

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

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