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