The notion that group theory captures the idea of "symmetry" derives from the notion of the symmetric group, and the very important theorem due to Cayley that every group is a subgroup of a symmetric group.
Let be a set. A bijection is a permutation of . Write for the set of permutations of the set (so its elements are functions).
Then is a group under the operation of composition of functions; it is the symmetric group on . (It is also written , for the automorphism group.)
We write for , the symmetric group on elements.
We can represent a permutation of in two different ways, each of which is useful in different situations.
Let , so is a function . Then we write for . This has the advantage that it is immediately clear where every element goes, but the disadvantage that it is quite hard to see the properties of an element when it is written in double-row notation (for example, " cycles round five elements" is hard to spot at a glance), and it is not very compact.
Cycle notation is a different notation, which has the advantage that it is easy to determine an element's order and to get a general sense of what the element does. Every element of can be expressed in (disjoint) cycle notation in an essentially unique way.
It is a useful fact that every permutation in a (finite) symmetric group may be expressed as a product of transpositions.
Those are the only two abelian symmetric groups. Indeed, in cycle notation, and do not commute in for , because while .
A very important (and rather basic) result is Cayley's Theorem, which states the link between group theory and symmetry.
The alternating group is defined as the collection of elements of which can be made by an even number of transpositions. This does form a group (proof).