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A shortest conjugacy representative is a word having the fewest written letters among all words representing elements of a fixed conjugacy class. This minimum equals the least group word length of an element in the class; a longer spelling of that element is not itself a shortest representative. Such a word exists because lengths are nonnegative integers. It is freely and cyclically reduced: free cancellation shortens the same representative, while removing mutually inverse first and last letters shortens a conjugate. Every cyclic rotation has the same minimal length.

The word length $|g{|}_S$ is the least number of letters from a symmetric generating set of a group $S$ whose product is $g$. The empty word has length zero. It obeys $|gh{|}_S\le |g{|}_S+|h{|}_S$, $|g^{-1}{|}_S=|g{|}_S$ and $d_S(g,h)=|g^{-1}h{|}_S$. The generating set may be infinite; finiteness is needed only for additional conclusions such as finite metric balls.