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. For full proof, refer: Isolator of subgroup is subgroup in nilpotent group

References

Textbook references

• Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, ^{More info}, Page 50, Section 2.6 (formal definition)