In the mathematical discipline of 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.
More specifically, G is isomorphic to a subgroup of the symmetric group
Sym
(
G
)
{\displaystyle \operatorname {Sym} (G)}
whose elements are the permutations of the underlying set of G.
Explicitly,
for each
g
∈
G
{\displaystyle g\in G}
, the left-multiplication-by-g map
ℓ
g
:
G
→
G
{\displaystyle \ell _{g}\colon G\to G}
sending each element x to gx is a permutation of G, and
the map
G
→
Sym
(
G
)
{\displaystyle G\to \operatorname {Sym} (G)}
sending each element g to
ℓ
g
{\displaystyle \ell _{g}}
is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of
Sym
(
G
)
{\displaystyle \operatorname {Sym} (G)}
.
The homomorphism
G
→
Sym
(
G
)
{\displaystyle G\to \operatorname {Sym} (G)}
can also be understood as arising from the left translation action of G on the underlying set G.
When G is finite,
Sym
(
G
)
{\displaystyle \operatorname {Sym} (G)}
is finite too. 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
S
n
{\displaystyle S_{n}}
. But G might also be isomorphic to a subgroup of a smaller symmetric group,
S
m
{\displaystyle S_{m}}
for some
m
<
n
{\displaystyle m<n}
; for instance, the order 6 group
G
=
S
3
{\displaystyle G=S_{3}}
is not only isomorphic to a subgroup of
S
6
{\displaystyle S_{6}}
, but also (trivially) isomorphic to a subgroup of
S
3
{\displaystyle S_{3}}
. The problem of finding the minimal-order symmetric group into which a given group G embeds is rather difficult.
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".
When G is infinite,
Sym
(
G
)
{\displaystyle \operatorname {Sym} (G)}
is infinite, but Cayley's theorem still applies.
Contents
History
When Cayley (1854) introduced what are now called groups, the modern definitions did not exist, and it was not immediately clear that this was equivalent to what were then called 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's original 1854 paper,
showed that the correspondence in the theorem is one-to-one, but he did not 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.
Background
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
Sym
(
A
)
{\displaystyle \operatorname {Sym} (A)}
.
In particular, taking A to be the underlying set of a group G produces a symmetric group denoted
Sym
(
G
)
{\displaystyle \operatorname {Sym} (G)}
.
Proof of the theorem
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 also an 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)):
(
f
g
⋅
f
h
)
(
x
)
=
f
g
(
f
h
(
x
)
)
Alternative setting of proof
An alternative setting uses the language of group actions. We consider the group
G
{\displaystyle G}
as acting on itself by left multiplication, i.e.
g
⋅
x
=
g
x
{\displaystyle g\cdot x=gx}
, which has a permutation representation, say
ϕ
:
G
→
S
y
m
(
G
)
{\displaystyle \phi :G\to \mathrm {Sym} (G)}
.
The representation is faithful if
ϕ
{\displaystyle \phi }
is injective, that is, if the kernel of
ϕ
{\displaystyle \phi }
Remarks on the regular group representation
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
Z
2
=
{
0
,
1
}
{\displaystyle \mathbb {Z} _{2}=\{0,1\}}
with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12) (see cycle notation). E.g. 0 +1 = 1 and 1+1 = 0, so
1
↦
0
{\textstyle 1\mapsto 0}
and
0
↦
1
,
{\textstyle 0\mapsto 1,}
as they would under a permutation.
Z
3
=
{
0
,
1
,
2
}
More general statement
Theorem:
Let G be a group, and let H be a subgroup.
Let
G
/
H
{\displaystyle 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
{\displaystyle G/N}
is isomorphic to a subgroup of
Sym
(
G
/
H
)
{\displaystyle \operatorname {Sym} (G/H)}
.
The special case
H
=
1
{\displaystyle H=1}
is Cayley's original theorem.