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This question is a little tricky (for me, at least), since in the textbook the proof of

Theorem 5: Every subgroup of a free group is free.

is not yet provided (even though I've seen such proofs in Magnus et al.'s "Combinatorial Group Theory [...]," in Lyndon & Schupp's "Combinatorial Group Theory," and in Johnson's "Presentation$\color{red}{s}$ of Groups").

Here is the exercise:

Let $\Xi$ be a set of non-commuting variables. Consider the group of units of the ring of formal power series, with integer coefficients, in the non-commuting variables coming from $\Xi$. Prove that

$$F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$$

is free on $\{1+\xi\mid \xi\in\Xi\}$. (This is a theorem of W. Magnus.)


Before this exercise is the following

Hint:

Before starting out on the proof of Theorem 5, we give some examples of some groups which turn out to be free. The proofs require the criterion (iv), of Theorem 1 of Chapter 1. (Emphasis added.)

Okay, so here is that criterion:

Theorem 1: (iv) Let $G$ be a group and suppose that $X$ generates $G$. If every reduced $X$-word is different from $1$, then $G$ is a free group, freely generated by $X$.

Notation: Let $X\subseteq G$. The subgroup of $G$ generated by $X$ is denoted $\operatorname{gp}(X)$.

This notation is so that such subgroups are not confused with presentations.


Thoughts:

This is a question I think I should be able to do myself, especially since it seems to be a matter of letting $x_\xi:=1+\xi$ - or something to that effect - as there are no restrictions on the elements of $\Xi$. But then I suspect I'd be using Theorem 5, which would result in an annoying anachronism in my understanding of the material in the book.

Please help :)

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  • $\begingroup$ I'm a little unclear on what you want. Are you asking how to prove "the exercise" using Theorem 1? $\endgroup$ – Lee Mosher Apr 15 at 11:43
  • $\begingroup$ Yes, @LeeMosher. I'm sorry it's unclear. Please let me know how I can improve the question. $\endgroup$ – Shaun Apr 15 at 11:44
  • $\begingroup$ Writing an actual question somewhere, with a question mark, would help. $\endgroup$ – Lee Mosher Apr 15 at 11:45
  • $\begingroup$ Is that any better, @LeeMosher? $\endgroup$ – Shaun Apr 15 at 11:47
  • $\begingroup$ There's still no question mark, but clear enough from the comments, I suppose. $\endgroup$ – Lee Mosher Apr 15 at 11:49
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As I understand it, your job is to apply Theorem 1, not Theorem 5.

So to start, you should take a sequence of elements $\xi_1,...,\xi_n \in \Xi$ and a sequence of exponents $\epsilon_1,...,\epsilon_n \in \{-1,+1\}$, and you should assume that for all $i=1,...,n-1$ if $\xi_i = \xi_{i+1}$ then $\epsilon_i = \epsilon_{i+1}$. Using those assumptions, your job is then to prove that product $$(1 + \xi_1)^{\epsilon_1}\cdots(1 + \xi_n)^{\epsilon_n} \quad (*) $$ is not the identity element of the ring of formal power series. The assumptions were made in order to express that the product $(*)$ is a reduced word in the set of generators $X = \{1 + \xi \mid \xi \in \Xi\}$.

Can you take it from here?

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