Powering threshold

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