= p-rank of a group
{title2=$d_p(G)$}
For a <finitely generated group>, let $G^p$ denote the subgroup generated by all $p$th powers. Its p-rank is
$$
d_p(G)=\dim_{\mathbb F_p}\bigl(G/[G,G]G^p\bigr).
$$
This is the dimension of the largest quotient that is an <elementary abelian group> of exponent $p$. In a <group presentation> with $d$ generators and $s$ relators that are not $p$th powers, reduction of the exponent sums modulo $p$ gives at most $s$ linear constraints. Hence $d_p(G)\geq d-s\geq\operatorname{def}_p(\mathcal P)$. In particular a presentation of p-deficiency at least one yields a surjection onto $C_p$.
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