In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every groupG is isomorphic to a subgroup of the symmetric group acting on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a set G is any bijective function taking G onto G. The set of all permutations of G forms a group under function composition, called the symmetric group onG, and written as Sym. Cayley's theorem puts all groups on the same footing, by considering any group as a permutation group of some underlying set. Thus, theorems that are true for subgroups of permutation groups are true for groups in general. Nevertheless, Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups". The regular action used in the standard proof of Cayley's theorem does not produce the representation of G in a minimal-order permutation group. For example,, itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of . The problem of finding an embedding of a group in a minimal-order symmetric group is rather difficult.
History
While it seems elementary enough, at the time the modern definitions didn't exist, and when Cayley introduced what are now called groups it wasn't immediately clear that this was equivalent to the previously known groups, which are now called permutation groups. Cayley's theorem unifies the two. Although Burnside attributes the theorem to Jordan, Eric Nummela nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper, showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism. However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so. The theorem was later published by Walther Dyck in 1882 and is attributed to Dyck in the first edition of Burnside's book.
Proof of the theorem
If g is any element of a group G with operation ∗, consider the function, defined by. By the existence of inverses, this function has a two-sided inverse,. So multiplication by g acts as a bijective function. Thus, fg is a permutation of G, and so is a member of Sym. The set is a subgroup of Sym that is isomorphic to G. The fastest way to establish this is to consider the function with for every g in G. T is a group homomorphism because : for all x in G, and hence: The homomorphism T is injective since implies that for all x in G, and taking x to be the identity elemente of G yields, i.e. the kernel is trivial. Alternatively, T is also injective since implies that . Thus G is isomorphic to the image ofT, which is the subgroup K. T is sometimes called the regular representation ofG.
Alternative setting of proof
An alternative setting uses the language of group actions. We consider the group as a G-set, which can be shown to have permutation representation, say. Firstly, suppose with. Then the group action is by classification of G-orbits. Now, the representation is faithful if is injective, that is, if the kernel of is trivial. Suppose Then, by the equivalence of the permutation representation and the group action. But since, and thus is trivial. Then and thus the result follows by use of the first isomorphism theorem.
The identity element of the group corresponds to the identity permutation. All other group elements correspond to derangements: permutations that do not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation that consists of cycles all of the same length: this length is the order of that element. The elements in each cycle form a right coset of the subgroup generated by the element.
Examples of the regular group representation
Z2 = with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation. E.g. 0 +1 = 1 and 1+1 = 0, so 1 -> 0 and 0 -> 1, as they would under a permutation. Z3 = with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation, and group element 2 to permutation. E.g. 1 + 1 = 2 corresponds to =. Z4 = with addition modulo 4; the elements correspond to e,,,. The elements of Klein four-group correspond to e,,, and. S3 is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements, and the latter is how it is realized by its regular representation.
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More general statement of the theorem
A more general statement of Cayley's theorem consist of considering the core of an arbitrary group. In general if is a group and is a subgroup with, then is isomorphic to a subgroup of. In particular if is a finite group and we set then we get the classic result.
