# Conjugation family

From Groupprops

## Definition

Let be a finite group and be a -Sylow subgroup of , for some prime . A **conjugation family** for in is a family of subgroups of , such that the following holds.

### In terms of right actions

Whenever and are subsets of such that for some :

Then, and are -conjugate via . In other words, we can find elements , and subgroups such that:

- The subgroup generated by is in
- Each is in the normalizer of i.e.
- We have for any :

### In terms of left actions

Whenever and are subsets of such that for some :

Then, and are -conjugate via . In other words, we can find elements , and subgroups such that:

- The subgroup generated by is in
- Each is in the normalizer of i.e.
- We have for any :