Every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer
A related survey article is: finding conjugate subgroups
Suppose is a finite group and is a -Sylow subgroup of . Suppose is a subgroup contained inside . Then, there exists a subgroup of , such that and are conjugate subgroups inside , and such that is a Sylow subgroup of .
Given: A finite group , a -Sylow subgroup , a subgroup contained inside .
To prove: There exists a subgroup of such that and are conjugate subgroups inside , and such that ,math>N_P(H)</math> is -Sylow inside .
- Let be a -Sylow subgroup of . Such a exists by fact (1) in the subgroup .
- Let be such that . Note that such a exists by fact (2) in the group .
- Let . Then, is a ,math>p</math>-Sylow subgroup of . By construction, , so a -Sylow subgroup of is contained in . Thus, is -Sylow in .