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


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.


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 :)

  • $\begingroup$ 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

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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