History[edit]

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]

Background[edit]

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 $\operatorname {Sym} (A)$ .^[13] In particular, taking $A$ to be the underlying set of a group $G$ produces a symmetric group denoted $\operatorname {Sym} (G)$ .

Remarks on the regular group representation[edit]

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.

More general statement[edit]

Theorem: Let $G$ be a group, and let $H$ be a subgroup. Let $G/H$ 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 $G/N$ is isomorphic to a subgroup of $\operatorname {Sym} (G/H)$ .

The special case $H=1$ is Cayley's original theorem.