Torsion-free threshold

For a group
Suppose $$G$$ is a group. The torsion-free threshold for $$G$$ is the largest positive integer $$m$$ such that $$G$$ is $$p$$-torsion-free for all primes $$p \le m$$. In other words, for any prime $$p \le m$$, $$G$$ lacks any element of order $$p$$.

Note that if $$G$$ is a torsion-free group, we say that it has a torsion-free threshold of $$\infty$$.

For a non-associative ring
For a non-associative ring, the torsion-free threshold is defined as the torsion-free 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 torsion-free threshold for this sequence is the largest positive integer $$m$$ such that, for $$i \in \{ 1,2,\dots,m\}$$, the group $$G_i$$ is $$p$$-torsion-free for all primes $$p$$ 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 torsion-free threshold of $$\infty$$.

Related notions

 * Lower central series torsion-free threshold