Word

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.

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 $$G$$.