Filtered power automorphism

Definition

Suppose $G$ is a nilpotent group. Consider a central series of $G$ of the form:

$G=H_{1}\geq H_{2}\geq H_{3}\geq \dots \geq H_{n}=1$

A filtered power automorphism of $G$ corresponding to a rational number $r$ is an automorphism $\sigma$ of $G$ such that the following hold:

• $\sigma (H_{i})=H_{i}$ for each $i$.
• Each of the quotient groups $H_{i}/H_{i+1}$ is powered over all primes dividing the numerator or denominator of $r$.
• The induced automorphism by $\sigma$ on the quotient group $H_{i}/H_{i+1}$ is the powering map by $r^{i}$.