Uniform pro-p group
= Uniform pro-p group
A finitely generated <powerful p-group> in the category of <pro-p groups> is uniform if all successive indices in its <lower p-series> are equal. Equivalently it is torsion-free. If its generator number is $d$, the successive indices are $p^d$, the power map carries $G^{p^n}$ bijectively onto $G^{p^{n+1}}$, and an ordered minimal generating set gives unique coordinates $g_1^{\lambda_1}\cdots g_d^{\lambda_d}$ with $\lambda_i$ in the <p-adic integers>. For $p=2$, powerfulness means $[G,G]\leq G^4$.