Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 4 / 4 / a

In the uniform pro-p group $(L,\ast )$, the $p^n$-th power of $X$ is exactly $p^{n}X$, and its unique $p^n$-th root in the appropriate layer is division by $p^n$. The intrinsic addition is thereforeThe linear term is exact; every non-linear term tends to zero by the convergence estimates above. The intrinsic Lie bracket issince the leading term is $p^{2n}[X,Y]$ and the terms of degree at least three vanish after rescaling. Finally the intrinsic scalar operation sends $(\lambda ,X)$ to $\lambda X$. All underlying sets are identical, and all three operations agree. Thus