In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group.
[1] More specifically, G is isomorphic to a subgroup of the symmetric group
whose elements are the permutations of the underlying set of G. Explicitly, The homomorphism
can also be understood as arising from the left translation action of G on the underlying set G.[2] When G is finite,
The proof of Cayley's theorem in this case shows that if G is a finite group of order n, then G is isomorphic to a subgroup of the standard symmetric group
But G might also be isomorphic to a subgroup of a smaller symmetric group,
[3] The problem of finding the minimal-order symmetric group into which a given group G embeds is rather difficult.
[4][5] 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".
is infinite, but Cayley's theorem still applies.
While it seems elementary enough, at the time the modern definitions did not exist, and when Cayley introduced what are now called groups it was not 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[7] attributes the theorem to Jordan,[8] Eric Nummela[9] nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate.
Cayley, in his original 1854 paper,[10] showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding).
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[11] and is attributed to Dyck in the first edition of Burnside's book.
[12] A permutation of a set A is a bijective function from A to A.
The set of all permutations of A forms a group under function composition, called the symmetric group on A, and written as
[13] In particular, taking A to be the underlying set of a group G produces a symmetric group denoted
So multiplication by g acts as a bijective function.
Thus, fg is a permutation of G, and so is a member of Sym(G).
The set K = {fg : g ∈ G} is a subgroup of Sym(G) that is isomorphic to G. The fastest way to establish this is to consider the function T : G → Sym(G) with T(g) = fg for every g in G. T is a group homomorphism because (using · to denote composition in Sym(G)): for all x in G, and hence: The homomorphism T is injective since T(g) = idG (the identity element of Sym(G)) implies that g ∗ x = x for all x in G, and taking x to be the identity element e of G yields g = g ∗ e = e, i.e. the kernel is trivial.
Alternatively, T is also injective since g ∗ x = g′ ∗ x implies that g = g′ (because every group is cancellative).
Thus G is isomorphic to the image of T, which is the subgroup K. T is sometimes called the regular representation of G. An alternative setting uses the language of group actions.
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.
with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12) (see cycle notation).
with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132).
with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).
The elements of Klein four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).
be the set of left cosets of H in G. Let N be the normal core of H in G, defined to be the intersection of the conjugates of H in G. Then the quotient group