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Set $P_0=G$ and $P_{i+1}=P_i^p$. The permitted power-embedding result, applied repeatedly, gives $[P_i,G]\le P_{i+1}$, and each $P_i$ is a powerful p-group. These subgroups eventually reach $1$: if $P_i\ne 1$, its Frattini subgroup $P_i^p$ is proper. In fact these are the lower p-series layers, with their indices shifted by one.

For $x\in G$ and $y\in P_i$, the commutator $[y,x]$ lies in $P_{i+1}$, is central modulo $P_{i+2}$, and has exponent dividing $p$ there. The class-two collection calculation from part (iv) therefore givesApplying part (iv) inside $P_i$, whose third lower-series term is contained in $P_{i+2}$, shows that the map $y\mapsto y^p$ is onto $P_{i+1}/P_{i+2}$.

Given $a\in P_1$, first choose $x$ with $x^p\equiv a(\mathrm{mod}{P}_2)$. Suppose $x^p\equiv a(\mathrm{mod}{P}_{i+1})$. Choose $y\in P_i$ with $y^p\equiv x^{-p}a(\mathrm{mod}{P}_{i+2})$. The displayed collection congruence then gives $(xy{)}^p\equiv a(\mathrm{mod}{P}_{i+2})$. Replacing $x$ by $xy$ improves the approximation by one layer. Since the filtration on a group terminates, a finite sequence of corrections gives $x^p=a$ exactly. The reverse inclusion is the definition of $G^p$. Hence

A filtration on a group is a function $\omega :G\to \mathbb{R}\cup \{\infty \}$ satisfyingIt is a p-valuation when it is separated and, for $x\ne 1$,

Let $p$ be odd and letFor $g=1+A\ne 1$, define $\omega (g)={\min }_{i,j}{v}_p(A_{ij})$. Matrix multiplication and the identitygive the two filtration inequalities. Since $\omega (g)\ge 1>1/(p-1)$, only the p-power condition remains. The binomial theorem givesThe first term has valuation $1+\omega (g)$, while every other term has strictly larger valuation because $p$ is odd and $\omega (g)\ge 1$. Hence $\omega (g^p)=\omega (g)+1$, so this is a p-valuation.

Finally, if ${\omega }_1,{\omega }_2$ are p-valuations, put $\omega =\min ({\omega }_1,{\omega }_2)$. Taking minima preserves both filtration inequalities and the strict lower bound, whileIf $\omega (g)=\infty$, both original valuations force $g=1$. Therefore the pointwise minimum of two p-valuations is again a p-valuation.

For a prime number $p$, a p-valuation is a separated filtration on a group such that every $x\ne 1$ satisfiesA group equipped with one is called a p-valued group.