# Word

## Definition

### In the context of groups

A word in the context of groups is a formal expression in terms of a bunch of symbols (called letters) that is written as a string in terms of those symbols and their inverses. Suppose the word uses $n$ letters. For any group $G$, the word defines a set map (called the word map) from $G^n$ to $G$ by evaluation of the expression upon plugging in the values for the letters.

For instance, the following are all words in two letters $x_1,x_2$:

$x_1, x_2, x_1x_2, x_1x_2x_1^{-1}, x_2x_1x_2, x_1x_2x_1^{-1}x_2^{-1}, x_1x_1x_2$

The corresponding set map for the word $x_2x_1x_2$ is the map:

$G \times G \to G$

given by:

$(x_1,x_2) \mapsto x_2x_1x_2$

We are allowed to simplify the word using the general identities in groups, so, for instance, the word $x_1x_1^{-1}x_2$ is treated as equivalent to the word $x_2$. The word may be written in somewhat shorter form using the notational shortcuts for groups. For instance, the word $x_1x_1x_1x_2$ may be written in shorthand as $x_1^3x_2$. Similarly, $x_1x_2^{-1}x_2^{-1}x_2^{-1}x_1x_2$ can be written as $x_1x_2^{-2}x_1x_2$.

The empty word is taken to stand for the identity element.

### In the context of monoids

A word in the context of groups is a formal expression in terms of a bunch of symbols (called letters) that is written as a string in terms of those symbols. The key difference from words in the context of groups is that we are not allowed to use the inverse operation when constructing the word. Power notation can be used for positive powers just as in the context of groups, and the empty word stands for the identity element.

## Terminology

### Words versus their values

Although the term word should correctly be used only for the formal expression, it is sometimes used for the value taken by the formal expression for a particular choice of letter values in a particular group, i.e., the image of a particular tuple under the word map.

### Satisfaction of a word

For a word $w$ in $n$ letters and a group $G$, we sometimes say that a tuple $(g_1,g_2,\dots,g_n)$ satisfies $w$ if $w(g_1,g_2,\dots,g_n)$ is the identity element of $G$.

## Related notions

Term How it is related to words
Verbal subgroup This is a subgroup that can be expressed precisely as the union of the images of a bunch of word maps. Equivalently, it is a subgroup that can be expressed precisely as the subgroup generated by the union of the images of a bunch of word maps.
Monomial map This is a map from a group to itself obtained by starting with a word map and fixing all but one of the letters, with the unfixed letter the input of the map. Inner automorphisms, left multiplications, and right multiplications are all examples of monomial maps.
Homomorphism of groups Ostensibly, homomorphisms are set maps between groups that commute with the group operations. But it also follows from this that any homomorphism between groups commutes with any word map.
Generating set of a group This is a subset such that every element of the group can be expressed as a word in the generating set, i.e., as the image of the generating set under a word map.
Universal power map This is a word map with one letter, something of the form $g \mapsto g^n$.
Commutator This refers to a particular word in two letters that is the identity element if and only if the elements commute. $x_1x_2x_1^{-1}x_2^{-1}$ (left convention) or $x_1^{-1}x_2^{-1}x_1x_2$ (right convention).