Isolator

From Groupprops
Jump to: navigation, search

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