Isolator

Definition
Suppose $$G$$ is a group (not necessarily finite), $$S$$ is a subset of $$G$$, and $$\pi$$ is a set of primes. Then, the isolator of $$S$$ at the set $$\pi$$, denoted $$I_\pi(S)$$, is defined as:

$$I_\pi(S) := \{ x \in G \mid \exists n \in \langle \pi \rangle, x^n \in S \rangle$$

where $$n \in \langle \pi \rangle$$ means that $$n$$ is a $$\pi$$-number, i.e. all prime divisors of $$n$$ are in $$\pi$$.

Facts
In a nilpotent group, the isolator of any subgroup, with respect to any set of primes, is nilpotent.

Textbook references

 * , Page 50, Section 2.6 (formal definition)