Powering threshold

Definition

For a group

Suppose $G$ is a group. The powering threshold or unique divisibility threshold for $G$ is the largest positive integer $m$ such that $G$ is powered for all the primes less than or equal to $m$ .

Note that if $G$ is a rationally powered group, we say that it has a powering threshold of $\infty$ .

Note also that the powering threshold of a group is always one less than a prime number. Explicitly, if $p$ is the smallest prime such that $G$ is not $p$ -powered, the powering threshold is $p - 1$ .

For a non-associative ring

For a non-associative ring, the powering threshold is defined as the powering threshold of the additive group of the ring.

For a sequence of groups

Consider a sequence of groups $G_{1}, G_{2}, \dots, G_{n}, \dots$ . The powering threshold for this sequence is the largest positive integer $m$ such that, for $i \in {1, 2, \dots, m}$ , the group $G_{i}$ is powered for all the primes less than or equal to $i$ .

Note that if the condition holds for all positive integers $m$ (so there is no largest), we say that the sequence has a powering threshold of $\infty$ .

Related notions

Lower central series powering threshold can be defined for a group or for a Lie ring.
Powering threshold for an endomorphism of a group