= Index-p rewriting bound for p-deficiency
If $H$ is a <normal subgroup> of index $p$ in a group presented by $\mathcal P$, the <Reidemeister–Schreier theorem> produces a presentation $\mathcal Q$ with
$$
\operatorname{def}_p(\mathcal Q)-1\geq p\bigl(\operatorname{def}_p(\mathcal P)-1\bigr).
$$
The preimage of $H$ in the <free group> has rank $1+p(d-1)$. A relator $w^{p^k}$ produces at most $p$ conjugate relators of weight $p^{-k}$ if its root lies in that preimage. If the root lies outside, $k\geq1$, and the coset conjugates are redundant up to conjugation; one relator of weight at most $p^{-(k-1)}$ suffices. Thus the total weighted relator cost grows by at most $p$. Convergent infinite sums are handled term by term.
Back to article page
