Commentary by Chris Pressey

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Group Theory

Intuition behind Group Action - Mathematics Stack Exchange

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Nielsen--Schreier theorem - Wikipedia

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The Nielsen–Schreier theorem states "Any subgroup of a free group is itself free." Suppose we turn that on its head: "Any overgroup of a non-free group is itself non-free". Are these two statements equivalent?

I first thought they're not equivalent in some subtle way, because the latter seems to have a trivial proof. But does it?

Suppose g is a non-free group. Then it contains 2 elements i, j that are equal for reasons other than the group axioms. Now embed g in an overgroup h. Then h contains i and j and they're still equal.

But why are they equal? Playing the skeptic, could they not have become equal due solely to the group axioms and the presence of the additional elements in h?

You have to show that that can't happen, and that's hard. In fact it's tempting to invoke the Nielsen–Schreier theorem - "That would mean h is free, and it has a non-free subgroup g, and we know from the Nielsen–Schreier theorem that can't happen."

So it does seem like these two ways of stating the Nielsen–Schreier theorem are in fact equivalent.

Bass--Serre theory - Wikipedia

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abstract algebra - Is there a generalization of the free group that includes infinitely long words? - Mathematics Stack Exchange

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algorithms - Is Group Theory useful in Computer Science in other areas but cryptography? - Computer Science Stack Exchange

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Galois theory is fundamental for abstract interpretation, but Galois theory isn't group theory.

Do all Noether theorems have a common mathematical structure?

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This is actually rather interesting, but it's way over my head I'm afraid. Physics is not my strong suit.