Odd-prime principal p-adic congruence subgroup is uniform ($U_m=1+p^m{M}_d({\mathbb{Z}}_p),[U_m:U_{m+1}]=p^{d^2}$)

For an odd prime $p$, the matrix logarithm and matrix exponential are inverse maps between $1+p{M}_d({\mathbb{Z}}_p)$ and $p{M}_d({\mathbb{Z}}_p)$. They imply that the $p$th-power map carries $U_m$ bijectively onto $U_{m+1}$. Direct multiplication gives $[U_i,U_j]\subseteq U_{i+j}$, so the group is a powerful pro-p group. The quotient $U_m/U_{m+1}$ is the additive matrix space over ${\mathbb{F}}_p$, with constant index $p^{d^2}$. The $d^2$ matrices $1+p{E}_{ij}$ topologically generate, by successive approximation through these quotients. The lower p-series is exactly $U_m$, so the group is a uniform pro-p group and a torsion-free group. At $p=2$ the first congruence subgroup contains $-I$ and is not uniform.