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Sylow's First Theorem

Sylow's First Theorem

Letbe a finite group of orderand letbe a prime dividingIfis the highest power ofdividingthenhas a subgroup of orderThis subgroup is called a Sylow– subgroup of

Sylow's First Theorem tells us thathas sungroups of orderand 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 thenhas a subgroup of orderIt is important to note that ifis a divisor  ofthen need not have a subgroup of orderFor instance,has order 12, 12 is divisible by 6, yethas no subgroups of order 6. (though by Sylow's First Theorem, it has subgroups of orderand 3).

Example: Letbe any group of order 20=2^2 *5. We define a group action onso that elements ofact on the setconsisting of all 4 – element subsets offor all

In order to verify that this is a group action we need to show

a)so that

b)

c)

a)SupposeIt must be less than 4 then sofor somebut thenThis is a contraction hence

b)for allhence

c)

Since we have a group action we may sayare the distinct orbits. By the partition equationand by the Orbit – Stabilizer Theorem,

4845 is not divisible by 2 so there is at least onenot divisible by 2. For thisthen is divisible by 4.

Suppose thatthenBy considering all the elementsfor we obtain the right cosetofAll elements of this coset are inhencehencebut right cosets have the same number of elements as the set which gives rise to them henceand so