p-valued group ($(G,\omega )$)

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 $x,y\in G$,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.^[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.