7
Permutations
The aim of this section is to discuss a large class of groups, called permutation groups.
The central result of this section is Cayley’s theorem, which states that actually any group
is contained in this class of group. Thus, permutation groups give a generic example of a
group.
7.1
Permutations
Definition 7.1. A permutation of a set is a bijection of this set onto itself.
Note that
(i) if α : X → X is a permutation, than the inverse function α−1 : X → X is also a
permutation;
(ii) if α : X → X and β : X → X are permutations, one can form a composition
α ◦ β : X → X, which is also a permutation.
Theorem 7.2. Let A be a nonempty set, and let S be the collection of all permutations of
A. Then S is a group under the composition ◦.
Proof. Clearly, ◦ defines a binary operation on S. We need to check the three axioms G1,
G2, G3 of the group.
(G1) We have seen in section 1 that composition is associative.
(G2) Consider the mapping ι : A → A such that for all x ∈ A, ι(x) = x. Then for any
α ∈ S one has α ◦ ι = ι ◦ α = α. Thus, ι is the identity element on S.
(G3) For any permutation α ∈ S, its inverse α−1 satisfies α−1 ◦ α = α ◦ α−1 = ι. Thus,
α−1 is the inverse of α with respect to the group operation.
Notation Below we will denote composition of permutations α and β by αβ rather
than α ◦ β.
Definition 7.3.
1. Let A be any set. The group of all permutations of A is called a
symmetric group on A and denoted by Sym(A).
2. Any subgroup of Sym(A) (including Sym(A) itself) is called a permutation group on
the set A.
3. If A is the finite set {1, 2, . . . , n} of n elements, then Sym(A) is commonly denoted
Sn . We have already dealt with the groups Sn .
Proposition 7.4. The order of the group Sn is n! = 1 · 2 · · · · · (n − 1) · n.
Proof. The problem of computing the number of elements in Sn is the same as the problem
of computing the number of different ways the integers 1, 2, . . . , n can be placed in the n
blanks indicated (using each integer just once):
1 2 ... n
.
...
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If we begin filling in these blanks from the left, there are n possibilities for the first blank.
Once that choice has been made, there remain n − 1 possibilities for the second blank.
Then there are n − 2 possibilities for the third blank, and so on. Thus, the total number of
possibilities is n!.
Example 7.5. Let us consider the group S2 . This group consists of 2 elements:
1 2
1 2
ι=
and α =
.
1 2
2 1
Note that α2 = ι. It is straightforward to see that the function f : Z2 → S2 , f (0) = ι,
f (1) = α is an isomorphism.
Example 7.6. Let us discuss in detail the structure of the group S3 . It consists of 3! = 6
elements:
1 2 3
1 2 3
1 2 3
, ρ1 =
, ρ2 =
,
ρ0 =
1 2 3
2 3 1
3 1 2
(7.1)
1 2 3
1 2 3
1 2 3
µ1 =
, µ2 =
, µ3 =
.
1 3 2
3 2 1
2 1 3
The element ρ0 is the identity in S3 . The Cayley table for S3 looks as follows:
ρ0
ρ1
ρ2
µ1
µ2
µ3
ρ0
ρ0
ρ1
ρ2
µ1
µ2
µ3
ρ1
ρ1
ρ2
ρ0
µ2
µ3
µ1
ρ2
ρ2
ρ0
ρ1
µ3
µ1
µ2
µ1
µ1
µ3
µ2
ρ0
ρ2
ρ1
µ2
µ2
µ1
µ3
ρ1
ρ0
ρ2
µ3
µ3
µ2
µ1
ρ2
ρ1
ρ0
The elements {ρ0 , ρ1 , ρ2 } form a subgroup of S3 . This is easy to see by considering all
products and inverses of the elements of this set. Note that this subgroup is abelian.
Moreover, it is generated by ρ1 , since ρ21 = ρ2 . Thus, this group is cyclic and isomorphic to
Z3 . There are three other subgroups in S3 . These are {ρ0 , µ1 }, {ρ0 , µ2 } and {ρ0 , µ3 }. Note
that each of these three groups is cyclic of the order 2, and thus isomorphic to Z2 .
Example 7.7. The group GL(n, R) can be considered a subgroup of all permutations of
Rn . Indeed, we know (see MAA242 Linear Algebra) that any n × n matrix defines a linear
map from Rn to Rn . If the matrix is non-singular (i.e., its determinant is non-zero), then
this linear map is a bijection.
7.2
Cayley’s Theorem
Theorem 7.8. Every group is isomorphic to a permutation group on a certain set.
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An outline of the proof is given in the exercises.
At a first glance, this theorem seems to be a tool to answer all questions about groups.
However, in practice examining all permutation groups on sets of all sizes would be a tremendous task. Even the finite permutation groups Sn get too complicated for a comprehensive
analysis as n gets big enough.
The importance of Cayley’s Theorem is ideological rather than practical. It shows that
it is possible to view the group theory as the study of permutation groups. In other words,
permutation groups are a universal model for all possible groups.
Cayley’s Theorem is an example of what is known as a representation theorem — it tells
you that every group can be represented as (i.e., is isomorphic to) something reasonably
concrete.
7.3
Isotropy groups
Example 7.9. Let us show that the group S4 has a subgroup H isomorphic to S3 . Consider
the set of all permutations σ ∈ S4 that do not move the element 4: σ(4) = 4. Let us denote
the union of these elements by H. These elements are
1 2 3 4
1 2 3 4
1 2 3 4
,
,
,
1 2 3 4
2 3 1 4
3 1 2 4
1 2 3 4
1 2 3 4
1 2 3 4
,
,
.
1 3 2 4
3 2 1 4
2 1 3 4
Let us compare the above list with (7.1). Clearly, if we ‘forget’ about the last column, the
elements of H are exactly the elements of the S3 group. It is also easy to see that the
presence of the right column does not affect multiplication of the permutations. So, we
have just established an isomorphism between H and the group S3 .
The following theorem makes our reasoning in the above example more rigorous.
Theorem 7.10. Let G be a permutation group on a set A. For any element x ∈ A, the set
Gx = {g ∈ G | g(x) = x}
is a subgroup of G.
Proof. Recall that Theorem 3.7 gives sufficient conditions for a subset of a group to be a
subgroup. We shall check the three conditions of this theorem.
1. We need to check that Gx is closed under the group operation. Let g1 , g2 ∈ Gx . Then
g1 g2 (x) = g1 (g2 (x)) = g1 (x) = x, and thus g1 g2 ∈ Gx .
2. We need to check that the identity is in Gx . But for the identity map ι one has
ι(y) = y for all y ∈ A, so this is true for x in particular.
3. We need to check that g ∈ Gx implies g −1 ∈ Gx . For g ∈ Gx one has x = ι(x) =
(g −1 g)(x) = g −1 (g(x)) = g −1 (x), and thus g −1 ∈ Gx .
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Definition 7.11. The subgroup Gx in Theorem 7.10 is the isotropy subgroup of x (another
term is a stabilizer of x).
Example 7.12. 1. In the Example 7.9, the subgroup H was the isotropy subgroup of the
element 4.
2. In the group S3 (see Example 7.6) the subgroups {ρ0 , µ1 }, {ρ0 , µ2 } and {ρ0 , µ3 } are
isotropy subgroups of the elements 1, 2 and 3 respectively.
Remark. As in the Example 7.9, it is easy to see that any isotropy subgroup of Sn is
isomorphic to Sn−1 .
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