Minimal cyclic power diagram and relator root

Use the symmetrised presentation of Sc toolkit symmetrised relators and pieces. Let $g\mathrm{\ne }1$ have finite order in the quotient. Choose a shortest freely reduced word $z$ among all words representing conjugates of $g$. Such lengths form a nonempty subset of the natural numbers, so a minimum is attained by The well-ordering principle. The length is positive since the empty word represents the identity. The word $z$ is cyclically reduced: if $z=au{a}^{-1}$, conjugating by $a^{-1}$ would give the shorter representative $u$.

Let $n$ be the least positive integer with $g^n=1$, using powers in Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$. The word $z^n$ is null. Its diagrams exist by Sc toolkit van kampen existence; choose one of least area, again using well-ordering. This is a minimal cyclic power diagram for this choice of $z$. Since $g\mathrm{\ne }1$, $n\ge 2$. No simultaneous choice for all conjugacy classes is required.

A relator root is a nonempty word $v$ which is not literally a proper power and for which a cyclic rotation $r$ of a defining relator satisfies $r=v^m$ literally, for some integer $m\ge 1$. The root is a word; its image in the quotient is a separate object. A shortest nonempty word whose positive power equals a given relator is a root: a proper-power decomposition would give a still shorter such word. Existence follows since the relator itself is a candidate. Literal powers and quotient-group powers must be distinguished.