Cayley's theorem

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G 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. In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G 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 on G, and written as Sym(G). Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) 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, S 3 {displaystyle S_{3}} , itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of S 6 {displaystyle S_{6}} (a group of order 720). The problem of finding an embedding of a group in a minimal-order symmetric group is rather more difficult. 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 (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 and is attributed to Dyck in the first edition of Burnside's book. If g is any element of a group G with operation ∗, consider the function fg : G → G, defined by fg(x) = g ∗ x. By the existence of inverses, this function has a two-sided inverse, f g − 1 {displaystyle f_{g^{-1}}} . 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)):