## Sylow's First Theorem

Sylow's First Theorem

Let be a finite group of order and let be a prime dividing If is the highest power of dividing then has a subgroup of order This subgroup is called a Sylow – subgroup of  Sylow's First Theorem tells us that has sungroups of order and 3.

In fact a p – group has a subgroup corresponding to every divisor of it's order, so a finite group has a subgroup corresponding to every prime power divisor of it's order, so that if divides then has a subgroup of order It is important to note that if is a divisor  of then need not have a subgroup of order For instance, has order 12, 12 is divisible by 6, yet has no subgroups of order 6. (though by Sylow's First Theorem, it has subgroups of order and 3).

Example: Let be any group of order 20=2^2 *5. We define a group action on so that elements of act on the set consisting of all 4 – element subsets of  for all In order to verify that this is a group action we need to show

a) so that b) c) a)Suppose It must be less than 4 then so for some but then This is a contraction hence b) for all hence c) Since we have a group action we may say are the distinct orbits. By the partition equation and by the Orbit – Stabilizer Theorem,  4845 is not divisible by 2 so there is at least one not divisible by 2. For this then is divisible by 4.

Suppose that then By considering all the elements for we obtain the right coset of All elements of this coset are in hence hence but right cosets have the same number of elements as the set which gives rise to them hence and so  