# Group Operations

## The Symmetric Group

A permutation of a set *Menge*, the German word for set) is a bijective function

As an example, consider the set *star* is fixed, but the triangle and the circle are *permutated*.

For a more concrete example, the *shape* or the *nature* of the objects in the set is not relevant, all that needs to be known is the number of objects in

Generally, as such permutations are bijective, they are invertible, with their inverse being a, probably different, permutation of

For a finite set

One can consider those permutations to be the symmetries of an object. Imagine a square rotated by *vertices* have *moved*, but the square still looks the same. Can you figure out all the eight symmetries of the square?

If you ever wanted to see the mathematical beauty of symmetry, think about visiting the Qalat Al-Hamra, or the Alhambra, in Grenada, Spain.

## Cayley’s Theorem

Arthur Cayley was a British mathematician, and probably the first man to clearly define the concept of a group. One of his famous theorems states that **every group is isomorphic to a group of permutations**.

Two *objects* are said to be isomorphic, if there exists an isomorphism, which is a bijective homomorphism (from the ancient Greek words *ὁμός (homós)*, which means *equal* or *similar*, and *μορφή (morphé)*, which means *form*), between the two objects. Homomorphisms are structure preservering maps that identify objects which, although looking differently, are essentially the same.

To prove the theorem, let

Furthermore, *permutation groups* and are denoted by

## Group Operations

With Cayley’s theorem, it is now clear how to define a group operation on a set: An operation of a group

Obviously, the following two properties hold:

, where is the unit element of . .

Conversely, a map

### The Orbit

Let

Clearly, all the orbits define a partition of

A group operation is called transitive, or G is said to act transitively on

### The Stabilizer

Let

Obviously, the kernel

A group operation is said to be faithful, if its kernel is trivial, i.e. if

## The Orbit-Stabilizer Theorem

The size of the orbit of an element

For a finite set

For a proof, we must define a bijective map

The first thing to check is whether

By the definition of

## The Orbit-Counting Theorem

In the case of a finite group operating on a finite set, a theorem, probably due to Ferdinand Georg Frobenius, allows us to count the number of orbits

In other words, the number of orbits under a group operation is the average number of points, or elements of

To prove this, we use the Orbit-Stabilizer Theorem:

Further, using the fact that

## Exercises

### Exercise 1

Show that

### Exercise 2

Prove that

### Exercise 3

Prove that the isotropy group is indeed a subgroup of

## References

### Literature

- Algebra, by Serge Lang
- Algebra I, by Rudolf Scharlau

### Art

- The Art of M.C. Escher
- Wikipedia