Powering threshold for an endomorphism of a group

Definition

Suppose $G$ is a group and $\sigma$ is an endomorphism of $G$. The powering threshold for $\sigma$ is defined as the powering threshold of the sequence of groups $G_{i}=\sigma ^{i}(G)$, i.e., the sequence:

$\sigma (G),\sigma ^{2}(G),\sigma ^{3}(G),\dots$

In other words, it is the largest positive integer $m$ such that $\sigma ^{i}(G)$ is powered for all primes less than or equal to $i$.

Note that if the condition that "$\sigma ^{i}(G)$ is powered for all primes less than or equal to $i$" holds for all $i$, the powering threshold is $\infty$, and such an endomorphism is termed an infinitely powered endomorphism.

Definition for rings

The definition above can be applied to any additive endomorphism of a non-associative ring.