= Word length
{title2=$|g|_S$}
The <word length> is the cost of building a <group element> from the allowed <generators of a group>. For a symmetric <generating set of a group> $S$ of a <group> $G$, define
$$
|g|_S=\min\{n\geq0:g=s_1\cdots s_n,\ s_i\in S\}.
$$
The empty <word over an alphabet> represents the <identity element> and has length zero. A representative with this least number of letters labels a shortest <graph path> from $1$ to $g$ in the <Cayley graph>. Thus <word length> minimizes over all representatives, rather than counting the letters in whichever expression we happened to write down.https://webusers.imj-prg.fr/~kathryn.mann/classes/GGRminicourse.pdf[Mann, Definition 1.1: word norm]{ref}
For the additive <integers> with $S=\{-1,1\}$, the expression $1+1-1$ has three letters but represents $1$, whose <word length> is one. With $T=\{-2,-1,1,2\}$, the <integer> $5$ has <word length> three, realized by $2+2+1$. The <Cayley graphs of the integers> visualize why these are minimal. A <freely reduced word> is always shortest in a <free group> with its free generators and their inverses; <free reduction> alone need not give a shortest representative in a general <group>.
Concatenating representatives and reversing their letters with <inverse elements> give
$$
|gh|_S\leq|g|_S+|h|_S,\qquad |g^{-1}|_S=|g|_S,\qquad |g|_S=d_S(1,g).
$$
These make <word length> a <length function>. A finite $S$ gives finite <word metric balls>, since only finitely many <words over an alphabet> have length at most a fixed radius. The definition itself also works for infinite symmetric <generating sets of a group>; the bounded-metric example in <equivalence of finite word metrics> shows what finiteness adds.
Back to article page
