Given a set $X$, we can make a new set $X^{-1}$ consisting of “formal inverses” of elements of $X$. That is, we create a set of new symbols, one for each element of $X$, which we denote $x^{-1}$; so $$X^{-1}=\{x^{-1}\mid x\in X\}$$

By this stage, we have not given any kind of meaning to these new symbols. Though we have named them suggestively as $x^{-1}$ and called them “inverses”, they are at this point just objects.

Now, we apply meaning to them, giving them the flavour of group inverses, by taking the union $X\cup X^{-1}$ and making finite “words” out of this combined “alphabet”.

A finite word over $X\cup X^{-1}$ consists of a list of symbols from $X\cup X^{-1}$. For example, if $X=\{1,2\}$ ^[1], then some words are:

• The empty word, which we commonly denote $\epsilon$

• $\left(1\right)$

• $\left(2\right)$

• $\left(2^{-1}\right)$

• $(1,2^{-1},2,1,1,1,2^{-1},1^{-1},1^{-1})$

For brevity, we usually write a word by just concatenating the “letters” from which it is made:

• The empty word, which we commonly denote $\epsilon$

• $1$

• $2$

• $2^{-1}$

• $12^{-1}21112^{-1}{1}^{-1}{1}^{-1}$

For even more brevity, we can group together successive instances of the same letter. This means we could also write the last word as $12^{-1}21^3{2}^{-1}{1}^{-2}$.

Now we come to the definition of a freely reduced word: it is a word which has no subsequence $r{r}^{-1}$ or $r^{-1}r$ for any $r\in X$.

Example

If $X=\{a,b,c\}$, then we might write $X^{-1}$ as $\{a^{-1},b^{-1},c^{-1}\}$ (or, indeed, as $\{x,y,z\}$, because there’s no meaning inherent in the $a^{-1}$ symbol so we might as well write it as $x$).

Then $X\cup X^{-1}=\{a,b,c,a^{-1},b^{-1},c^{-1}\}$, and some examples of words over $X\cup X^{-1}$ are:

• The empty word, which we commonly denote $\epsilon$

• $a$

• $aaaa$

• $b$

• $b^{-1}$

• $ab$

• $a{b}^{-1}cb{b}^{-1}{c}^{-1}$

• $a{a}^{-1}a{a}^{-1}$

Of these, all except the last two are freely reduced. However, $a{b}^{-1}cb{b}^{-1}{c}^{-1}$ contains the substring $b{b}^{-1}$, so it is not freely reduced; and $a{a}^{-1}a{a}^{-1}$ is not freely reduced (there are several ways to see this: it contains $a{a}^{-1}$ twice and $a^{-1}a$ once).

Alternative, more opaque, treatment which might help with one aspect

This chunk is designed to get you familiar with the idea that the symbols $a^{-1}$, $b^{-1}$ and so on in $X^{-1}$ don’t have any inherent meaning.

If we had (rather perversely) gone with $\{x,y,z\}$ as the corresponding “inverses” to $\{a,b,c\}$ (in that order), rather than $\{a^{-1},b^{-1},c^{-1}\}$ as our “inverses” ^[2], then the above words would look like:

• The empty word, which we commonly denote $\epsilon$

• $a$

• $aaaa$, which we might also write as $a^4$

• $b$

• $y$

• $ab$

• $aycbyz$

• $axax$

For the same reasons, all but the last two would be freely reduced. However, $aycbyz$ contains the substring $by$ so it is not freely reduced; and $axax$ is not freely reduced (there are several ways to see this: it contains $ax$ twice and $xa$ once).

Why are we interested in this?

We can use the freely reduced words to construct the free group on a given set $X$; this group has as its elements the freely reduced words over $X\cup X^{-1}$, and as its group operation “concatenation followed by free reduction” (that is, removal of pairs $r{r}^{-1}$ and $r^{-1}r$). ^[3] The notion of “freely reduced” basically tells us that $r{r}^{-1}$ is the identity for every letter $r\in X$, as is $r^{-1}r$; this cancellation of inverses is a property we very much want out of a group.

The free group is (in a certain well-defined sense from category theory^[4]) the purest way of making a group containing the elements $X$, but to make it, we need to throw in inverses for every element of $X$, and then make sure the inverses play nicely with the original elements (which we do by free reduction). That is why we need “freely-reducedness”.

1. ^︎

   Though in general $X$ need not be a set of numbers.

2. ^︎

   Which you should never do. It just makes things harder to read.

3. ^︎

   We make this construction properly rigorous, and check that it is indeed a group, on the Free group page.

4. ^︎

   See Free_group_functor_is_left_adjoint_to_forgetful for the rather advanced reason why.