Cayley's theorem

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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.[1] This can be understood as an example of the group action of G on the elements of G.[2]

A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G).[3]

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 which are true for subgroups of permutation groups are true for groups in general.

History

Although Burnside[4] attributes the theorem to Jordan,[5] Eric Nummela[6] nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,[7] showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

Proof of the theorem

Where g is any element of a group G with operation ∗, consider the function fg : GG, defined by fg(x) = gx. 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(G).

The set K = {fg : gG} 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 also injective since T(g) = idG (the identity element of Sym(G)) implies that gx = x for all x in G, and taking x to be the identity element e of G yields g = ge = e. Alternatively, T is also injective since, if gx = 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.

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 (also known as the orbit-stabilizer theorem).

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.

Remarks on the regular group representation

The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does 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 which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left coset of the subgroup generated by the element.

Examples of the regular group representation

Z2 = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12). E.g. 0 +1 = 1 and 1+1 = 0, so 1 -> 0 and 0 -> 1, as they would under a permutation.

Z3 = {0,1,2} 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). E.g. 1 + 1 = 2 corresponds to (123)(123)=(132).

Z4 = {0,1,2,3} 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).

S3 (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements:

* e a b c d f permutation
e e a b c d f e
a a e d f b c (12)(35)(46)
b b f e d c a (13)(26)(45)
c c d f e a b (14)(25)(36)
d d c a b f e (156)(243)
f f b c a e d (165)(234)

See also

Notes

  1. Jacobson (2009), p. 38.
  2. Jacobson (2009), p. 72, ex. 1.
  3. Jacobson (2009), p. 31.
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References

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