Sylow's Third Theorem
All the Sylow p subgroups (remember that if
is the highest power of
dividing
then the Sylow– subgroup of
is that subgroup which has order) of any groupare conjugate. Moreover if the number of Sylow– subgroups ofis
then
divides
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) and
divide 6.
Example: Example:
of order 8 has 5 subgroups of order 2:
The subgroups consist of one rotation group
of order 2 and four reflection groups {e,
There is only one Sylow 2 – subgroup,
Note that 1 divides
and 1 =1 (mod 8) so Sylow's Third Theorem is satisfied.