Sylow's Third Theorem

Sylow's Third Theorem

All the Sylow p subgroups (remember that ifis the highest power ofdividingthen the Sylow– subgroup ofis that subgroup which has order) of any groupare conjugate. Moreover if the number of Sylow– subgroups ofisthendivides

Example:has a unique Sylow 3 – subgroup

so is congruent to 1 (mod 3).

and 3 Sylow 2 - subgroups

still has one subgroup of order Sylow 3 – subgroup of order 3, but only one Sylow 2 – subgroup of order 2. Nevertheless, Sylow's third theorem is still satisfied, since(mod) 2) anddivide 6.

Example: Example:of order 8 has 5 subgroups of order 2:

The subgroups consist of one rotation groupof order 2 and four reflection groups {e,

There is only one Sylow 2 – subgroup,Note that 1 dividesand 1 =1 (mod 8) so Sylow's Third Theorem is satisfied.