Maths - Sylow Theory

Orbit

orb(s) - When a group G acts on a set S, the orbit of any s in S is the set of elements of S that G arrows can reach from s.

orb(s) = {φ(s) | φ∈G}

orb(s)

Stabiliser

stab(s) - The stabilizer of an element s in S is the set of group elements g that don't move s. A configuration s in S is called stable if no actions move s.

stab(s) = {φ∈G | φ(s)=s}

stab(s)

Example

In this example we take 6 permutations named g1 to g6:

g1
g2
g3
g1 permutation g2 permutation g3 permutation
g4
g5
g6
g4 permutation g5 permutation g6 permutation

we can calculate orb() and stab() examples as follows:

orb(s) stab(s) |orb(s)| · |stab(s)|
orb(1)={1,2,3} stab(1)={g1,g4} 6
orb(4)={4,5} stab(4)={g1,g2,g3} 6
orb(6)={6,7,8} stab(6)={g1,g4} 6

Orbit-Stabilizer Theorem

The size of an element's orbit times the size of its stabilizer is the size of the group.

|orb(s)| · |stab(s)| = |G|


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