= p-valued group
{title2=$(G,\omega)$}
In the p-adic Lie-theory convention, a p-valued group is a <group> $G$ equipped with a real-valued filtration $\omega$, with $\omega(e)=\infty$, such that for $x,y\in G$,
$$
\omega(xy^{-1})\geq\min(\omega(x),\omega(y)),\qquad \omega([x,y])\geq\omega(x)+\omega(y),
$$
and, for $x\ne e$, $1/(p-1)<\omega(x)<\infty$ and $\omega(x^p)=\omega(x)+1$. Here $p$ is prime and $[x,y]=x^{-1}y^{-1}xy$. The valuation induces a topology through neighborhoods where $\omega$ is large. Completeness or compactness is an additional hypothesis, not part of these axioms.<Some complements to the Lazard isomorphism by Annette Huber and others>[Definition2.1.1 and Proposition2.1.2, printed pp236–237]{ref}
For odd $p$, the additive group $p\mathbb Z_p$ with $\omega(x)=v_p(x)$ is an example: subtraction satisfies the non-Archimedean inequality, commutators vanish, and multiplying by $p$ raises valuation by one. For $p=2$, use $4\mathbb Z_2$ so the strict threshold exceeds one. The power axiom prevents a nonidentity element from having finite p-power order, and the commutator bound makes the associated graded object behave like a <graded Lie algebra>.
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