Let me describe how the free group on countably infinitely many generators embeds inside the free group on two generators:

First of all, let’s make the observation of Cayley’s theorem: Any group embeds into the “concrete” group of its own self-bijections. We can think of each group element as specifying the operation “Multiply by me”.

In particular, the free group on infinitely many generators can be thought of as a group of words, but it can also be thought of as a group of permutations on those words, with each generator representing the operation “Concatenate me to the word”. The group generated by all these concatenation operations is freely generated by them, being exactly isomorphic to the underlying free group of words they act on.

Now I want to consider another permutation of words:

We can imagine our countably infinitely many generators as arranged in a big infinite line, such that each generator has a successor and a predecessor. (As though the generators were in correspondence with the integers, say)

Now, we can permute words to “successors” or “predecessors” too, by just bumping up each instance of a generator (or its inverse) within them accordingly.

So, for example, if B is the successor of A, and C is the successor of B, and D is the successor of C, then the successor of the word BAC^{-1}ACAAB is CBD^{-1}BDBBC.

Now, pick any particular generator (let’s call it A like just before) and consider the group generated by two particular word-permutations: The concatenation with A permutation, and the successor permutation.

Note that concatenation with B is exactly the same as taking the predecessor of a word, then concatenating A, then taking the successor of the word.

Similarly, concatenation with C is exactly the same as taking the predecessor of a word, then concatenating B (as above), then taking the successor of the word.

And so on. So all the concatenation with a generator operations are within the group generated by “Concatenation with A” and successor.

That means a free group on infinitely many generators (the group of all concatenation permutations) is embedded within a group generated by two elements (the group of word-permutations generated by “Concatenation with A” and successor).

In abstract, this means A, Successor A Successor^{-1}, Successor^2 A Successor^{-2}, and so on, are all independent from each other and thus are the generators of a free group. This is true within the particular concrete group of permutations we have here (where A and Successor may, for all I’ve shown so far, have some relations with respect to each other…), but therefore also true within the free group on two generators called A and Successor.

TODO: Discuss the Nielsen–Schreier theorem. Discuss x-ly presented vs. x-ly generated vs. free on x generators.

TODO: Make another post on semidirect products. The way we embedded a group into its self-bijection (but not self-group-automorphism!) group via Cayley’s theorem, then augmented by further self-bijections which are in fact group automorphisms, amounts to the general semidirect product construction, though I’ve not seen it presented this way. We can also think of the general semiproduct construction as given by taking the coproduct of two groups G and H and adding a condition for what happens as one tries to commute g past h; instead of gh = hg, we get gh = h phi_h(g), say. Essentially, we turn an arbitrary automorphism into an inner automorphism. It can also be seen as an instance of the Grothendieck construction; see nLab.