Sc toolkit symmetrised relators and pieces

Let $X^{\pm 1}$ be an alphabet with formal inverses. Length $\mathrm{\mid }u\mathrm{\mid }$ means literal letter length. Fix a set $R$ of nonempty cyclically reduced words, closed under inverses and cyclic rotations, with duplicate words removed. The group is $G=⟨X\mid R⟩$ in Group presentation by generators and relations, and cyclic reduction has the convention of Cyclically reduced words. Neither $X$ nor $R$ must be finite.

A piece is a nonempty word $p$ that is an initial segment of two distinct words $r,s\in R$. The condition $C^{\mathrm{\prime }}(1\mathrm{/}6)$ says $\mathrm{\mid }p\mathrm{\mid }<\mathrm{\mid }r\mathrm{\mid }\mathrm{/}6$ for each such initial segment of each $r$. Rotating relators gives the identical bound for an overlap based anywhere on a relator. Equal rotations, including equal rotations of a proper power, are one word and do not create a piece.

Symmetrising a collection does not change its normal closure: if $r=uv$, then $vu=u^{-1}ru$ in the free group, and inverses of elements of a normal subgroup remain in it. Thus every added word belongs to the old normal closure; the original words are retained, giving the reverse inclusion. The empty set $R$ is permitted and satisfies the condition vacuously.

Touikan §3.5 uses occurrences and an irredundancy assumption. Here distinct full words control pieces, including for proper powers; the local proofs use precisely this convention.