A group action of on a set is the same as a homomorphism, where is the group of all bijections under composition.
Proof Given an action defined, for every , a map by . Then is a bijection because is its inverse. Therefore, we get a map Now check that is a homomorphism. Indeed, Conversely, given such , define by . This is a group action and the two constructions we defined are inverses to each other.
Proof Reflexivity follows from the fact that . Symmetry follows from the fact that if then . Transitivity follows from the fact that if and then .
Orbit and Stabiliser
Let be a group acting on a set and let . Then, the orbit of is the subset .
The stabilizer of in is the subgroup .
e.g. When acts on itself by conjugation, the orbits are called conjugacy classes, and the stabilizers are centralisers.
Classification of Actions
Transitive Action
A group action is transitive if it has only one orbit. This means for any two elements , there exists an element such that .
e.g. Let , then acts transitively on the left coset space by left multiplication, because any two cosets and are related by .
Free Action
A group action of on is said to be free if for some implies .
Faithful (Effective) Action
An action of on is said to be faithful if the only element of that acts as the identity on is the identity element of .
The Orbit-Stabilizer Theorem
Orbit-Stabilizer Theorem
Let be a group acting on a set . Let and let be the stabilizer of in and let ) be the orbit. Then the the map is a bijection. In particular, if is finite then . So the size of every orbit divides the order of the group.
Proof The map is surjective: an inverse image of is . The map is injective since: $$ gx = hx \iff h^{-1}gx=x \iff h^{-1}g\in G_x \iff gG_{x}=hG_{x}. $$$\square$
Class Equation
Let be a finite group, acting on a finite set . Then
Proof We claim that is a disjoint union of orbits. That is, and are either the same or disjoint. Suppose is nonempty and we pick some , then for some . It follows that , so . Similarly, . Hence . For every orbit we have by Orbit-Stabilizer Theorem.
e.g. If a finite group acts on itself by conjugation, then the orbits are the conjugacy classes and the stabilizers are the centralisers. The class equation is where are representatives of the non-trivial conjugacy classes.
-groups
-group
A finite group is called a -group if its order is a -power for some prime .
Proof Let be a finite −group and make it act on itself by conjugation. Observe that:By class equation, we know that . Note that each is a -power, if were trivial, then there is only one term , and all other terms are at least , so for some , which is not divisible by , a contradiction. Therefore, is non-trivial.
Proof A finite -group always has some central elements, and after removing them by quotienting, the quotient is still a smaller -group with more central elements. Eventually every element is accounted for by successive centers.
Proof We make induction on . If , there is nothing to prove. Assume that it is true for and we prove for the case . Note that is nilpotent, so is solvable. Then there exists such that and is abelian,
Cauchy’s Theorem
If is a prime dividing the order of a finite group then contains an element of order .
Proof Let . The group acts on by This is a well-defined group action because the multiplication of the RHS is Observe that is a fixed point (i.e., ) if and only if and . If there were no elements of order , then only , and all other will have . It follows that for some . This yields a contradiction because must divides .
Semidirect Products
Semidirect Product
Given groups and , and an -action on by automorphisms, say , the semidirect product is the set , with multiplication defined by:
e.g. If is the trivial action then this construction reduces to the usual direct product .