Cayley's theorem

The Cayley is a named after the English mathematician Arthur Cayley theorem from algebra. He says that one can realize each group as a subgroup of a symmetric group.

This result played for the development of group theory in the 19th century an important role, because it ensures that every abstract group is isomorphic to a concrete group of permutations. In other words, each group can be faithfully represented as a permutation. The Cayley thus forms a starting point for the representation theory, which examines a given group by exploiting their presentations to specific and well-understood groups.

Statement of the theorem

The Cayley says:

In more detail the following:

If the given group also finite, then one can choose a finite set this. More precisely: Is of order, then is isomorphic to a subgroup of

Applications

The practical significance of the set of Cayley is to represent any group as a subgroup of a specific group. As specific group is considered here is a symmetric group consisting of all bijective maps of a lot in itself. The link in the symmetric group is given by the sequential execution. Permutation groups are very practical in the sense that one can ( permutations ) write down conveniently and easily expect them their elements. This is especially useful in computer algebra.

On a theoretical level, the set of Cayley opens up the possibility of applying the theory of permutation groups on any group. One speaks of a permutation of the given group. In addition, there are other ways to represent groups in special form, for example as a matrix group, ie, as a subgroup of a linear group. This is called a linear representation, see the article representation theory ( group theory).

Proof of the theorem

Before the actual evidence it is worthwhile to illustrate the essential idea of a simple example. The following proof then formulated the observations made from only.

Introductory Example

Consider for illustration the Klein four-group, which we represent here by the quantity with the following truth table:

In the first line we see the permutation, and in the following lines the permutations. These permutations are different from each other, with the picture is therefore injective. It now expects directly after that a group homomorphism, ie satisfies for all. This follows quite generally from the group axioms, as we will show now.

General construction

Let be a group. As quantity we choose. For each group element, we define a mapping by. This mapping is called left multiplication by.

Since all elements are invertible in a group, therefore, each of the pictures is bijective. Thus we obtain by a group homomorphism. This homomorphism is injective: if, then, and therefore is especially true. This is an isomorphism between the group and the subgroup.

Comments

The above proof is based on with the observation that the left multiplication is a group action of the group on itself, viz. He then shows that each group induces a group homomorphism operation. In the specific case of the left multiplication is even injective, and is called the (left) regular representation.

The proof can be guided analogy, if the right multiplication is used instead of the left multiplication by the inverse. He then supplies may have a different subset of who is likewise isomorphic.

Minimal permutation representations

Instead of the amount used in the above proof, one can often find smaller quantities. For example, the proof is a representation of the alternating group of elements as a subset of, although the amount would be sufficient as the basic amount, because we have the inclusion.

For a given group can ask themselves at what level an injective group homomorphism exists (also called faithful permutation representation or embedding - see the events described in this section questions the article permutation ). The sentence makes it clear that this is always possible for any event. It is an interesting and sometimes difficult question of determining the minimal degree for which this is possible.

Interestingly, there are groups for which the regular representation is already minimal, ie. For such a group so there are embeddings only. For example, for each cyclic group of prime order, because no symmetric group contains an element of order ( Lagrange's theorem ). The same applies to any cyclic group whose order is a prime power is not symmetric group contains an element of order. ( This follows from the decomposition of a permutation into a product of disjoint Cycles. ) Also, the Klein four- group of order can not be in but in embed ( also according to the theorem of Lagrange ). A complete overview gives the following result:

For the following groups, the regular representation is already minimal, ie, there are embeddings only for:

In the cases (2) and (3) each embedded is conjugated to the regular representation.

Conversely, if the regular representation is minimal for a finite group, is a group selected from this list. For all other groups, ie, the degree of the set of Cayley even reduce.

History

The phrase is commonly Arthur Cayley attributed to the in 1854 formulated the basic idea in one of the first article of the group theory. However, William Burnside leads in his book on group theory returns the full evidence on Camille Jordan in 1870. Eric Nummela argues, however, that the usual description as a set of Cayley 's quite correct: Cayley had shown in his work of 1854 that the above figure in the symmetric group is injective, even if he did not explicitly shown that it is a group homomorphism.