Given a set , 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 $X^{- 1} = {x^{- 1} ∣ x \in X}$

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 $ε$

$(1)$

$(2)$

$(2^{- 1})$

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

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

The empty word, which we commonly denote $ε$

$2^{- 1}$

$1 2^{- 1} 2111 2^{- 1} 1^{- 1} 1^{- 1}$

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

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 $ε$

$a a a a$

$b^{- 1}$

$a b$

$a b^{- 1} c b b^{- 1} c^{- 1}$

$a a^{- 1} a a^{- 1}$

Of these, all except the last two are freely reduced. However, $a b^{- 1} c b b^{- 1} c^{- 1}$ 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 $ε$

$a a a a$ , which we might also write as $a^{4}$

$a b$

$a y c b y z$

$a x a x$

For the same reasons, all but the last two would be freely reduced. However, $a y c b y z$ contains the substring $b y$ so it is not freely reduced; and $a x a x$ is not freely reduced (there are several ways to see this: it contains $a x$ twice and $x a$ 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".

^{^︎}
Though in general $X$ need not be a set of numbers.
^{^︎}
Which you should never do. It just makes things harder to read.
^{^︎}
We make this construction properly rigorous, and check that it is indeed a group, on the Free group page.
^{^︎}
See Free_group_functor_is_left_adjoint_to_forgetful for the rather advanced reason why.

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Freely reduced word

Example

Why are we interested in this?