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I'm following the proof for this theorem in my textbook, and there is one part of it that I can't understand. Hopefully you can help me. Here is the part of the theorem and proof up to where I'm stuck:

Let ##N## be a normal subgroup of a group ##G##. Then every subgroup of the quotient group ##G/N## is of the form ##H/N## for some subgroup ##H## of ##G## with ##H\le G##.

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Proof: Let ##H^*## be a subgroup of ##G/N## so that it consists of a certain set ##\{hN\}## of left cosets of ##N## in ##G##. We define the subset ##\beta(H^*)## of ##G## to be ##\{g\in G:gN\in H^*\}##. Then ##\beta(H^*)## clearly contains N and is a subgroup of G:

[here follows demonstration that ##\beta(H^*)## is a subgroup of G]

The portion in red is what I'm having trouble with. I don't see why it's immediately clear that ##N## is contained within ##\beta(H^*)##.

Any help is appreciated.

I first thought that perhaps ##H^*## is supposed to correspond to the ##H## of the hypothesis, but then I realised that cannot be true, since the elements of ##\beta(H^*)## would be precisely those of ##H## and the properties would be trivial. In particular, proving that ##\beta(H^*)## is a subgroup of ##G## would be pointless.

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# Correspondence Theorem in Group Theory

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