Download 1 Groups

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Revision Notes on Group Theory
Introduction
By a group we mean a set together with some algebraic operation (such as addition
or multiplication of numbers) that satisfies certain rules. There are many examples of
groups in Mathematics, so it makes sense to understand their general theory, rather
than try to reprove things every time we come across a new example.
Common examples of groups include the set of integers together with addition,
the set of nonzero real numbers together with multiplication, the set of invertible
n × n matrices together with matrix multiplication.
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Binary Operations
The formal definition of a group uses the notion of a binary operation. A binary
operation ∗ on a set A is a map A × A → A, written (a, b) 7→ a ∗ b. Examples include
most of the standard arithmetic operations on the real or complex numbers, such as
addition (a + b), multiplication (a × b), subtraction (a − b). Other examples of binary
operations (on suitably defined sets) are exponentiation ab (on the set of positive
reals, for example), composition of functions, matrix addition and multiplication,
subtraction, vector addition, vector procuct of 3-dimensional vectors, and so on.
Definition A binary operation ∗ on a set A is commutative if a ∗ b = b ∗ a ∀a, b ∈ A.
Addition and multiplication of numbers is commutative, as is addition of matrices or vectors, union and intersection of sets, etc. Subtraction of numbers is not
commutative, nor is matrix multiplication.
Definition A binary operation ∗ on a set A is associative if a ∗ (b ∗ c) = (a ∗ b) ∗ c
∀a, b, c ∈ A.
Addition and multiplication (of numbers and matrices) are associative. Examples
of nonassociative binary operations are subtraction (of anything), exponentiation of
positive reals, and vector product.
Definition An identity for a binary operation ∗ on a set A is an element e ∈ A such
that e ∗ a = a = a ∗ e ∀a ∈ A.
Examples are 0 for addition of numbers, 1 for multiplication of numbers, the identity n × n matrix for matrix multiplication. Not all binary operations have identities,
however: an example is subtraction of numbers.
Definition Let ∗ be a binary operation on a set A and let a ∈ A. An inverse for a
(with respect to ∗) is an element b ∈ A such that a ∗ b and b ∗ a are identities for ∗.
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Thus for example −5 is an inverse for 5 with respect to addition of integers; 23 is
an inverse for 23 with respect to multiplication of positive real numbers. Other examples are matrix inverses (matrix multiplication) and appropriately defined inverse
functions (function composition).
Lemma If a binary operation on a set has an identity, then this identity is unique.
Proof. Suppose that e and f are both identities for a binary operation ∗ on a set A.
Then e = e ∗ f = f . The first equality holds because f is an identity. The second
holds because e is an identity.
Lemma If a ∈ A has an inverse with respect to an associative binary operation ∗ on
A, then the inverse is unique.
Proof. Suppose that b and c are both inverses for a. Then b = b∗(a∗c) = (b∗a)∗c = c.
The first equaliy holds because a ∗ c is an identity, the second because ∗ is associative,
and the third because b ∗ a is an identity.
In the last result, the associativity of ∗ is definitely used in the proof. In fact the
result is not in general true for nonassociative binary operations.
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Groups
We are now ready to give a definition of a group.
Definition A group is a nonempty set G together with a binary operation ∗ on G
which is associative and such that every element of G has an inverse. If the binary
operation ∗ is also commutative, then G is called a commutative, or abelian group
(after a 19th century Norwegian mathematician Niels Abel).
In particular, G has an identity, since there is at least one element a ∈ G, which
has an inverse b, and a ∗ b is an identity. By the above results the identity in G is
unique. We often denote it e = eG or 1 = 1G . The inverse of an element a is also
unique. We often denote it ā, or a−1 . If the binary operation on G is addition of
some sort, it is usual to denote it by +, the identity by 0 or 0G , and the inverse of an
elment a by −a.
Examples
1. Z, Q, R and C are all groups with respect to addition. In each case 0 is the
identity and the inverse of x is −x.
2. Any vector space V is a group with respect to vector addition. The identity is
the zero vector, and the inverse of v ∈ V is −v.
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3. The set Q∗ of nonzero rational numbers is a group with respect to multiplication.
The identity is 1, and the inverse of ab is ab . Similarly the sets R∗ and C∗ of
nonzero real and complex numbers, respectively, are groups with respect to
multiplication.
4. The set S 1 = {z ∈ C : |z| = 1} of complex numbers of modulus 1 is a group
with respect to multiplication of complex numbers.
5. The set of invertible n × n matrices forms a group with respect to matrix
multiplication. The identity element is the n × n identity matrix In .
6. Let X be a set. Then the set S(X) of all permutations of X, that is, bijective
maps X → X, forms a group with respect to composition of maps. The identity
map X → X is the identity element. This group is called the symmetric group
on X. In the particular case where X is the set {1, 2, . . . , n}, this group is
denoted Sn , and called the symmetric group of degree n.
7. Let n > 0 be an integer, and let Zn denote the set {0, 1, . . . , n − 1}. Define a
binary operation ∗ on Zn by a ∗ b = a + b if a + b < n, and a ∗ b = a + b − n
otherwise. Then Zn is an abelian group with respect to ∗, with identity 0. The
inverse of a > 0 in Zn is n − a (the inverse of 0 is 0). This group is called
the cyclic group of order n. The binary operation ∗ is usually denoted +, and
referred to as addition modulo n.
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Cayley tables
One way of describing a binary operation ∗ on a set G (provided G is not too big) is
to form a grid with rows and columns labelled by the elements of G, and enter the
element a ∗ b in the cell in row a and column b (for all a, b ∈ G). This is called a
multiplication table or a Cayley table or (in the case where (G, ∗) is a group) a group
table.
Example
+
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
This is the Cayley table for Z4 , the cyclic group of order 4.
Example
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∗
e
a
b
c
e
e
a
b
c
a
a
e
c
b
b c
b c
c b
e a
a e
This describes a binary operation on the set G = {e, a, b, c} with respect to which
G is a group.
Two groups G and H are said to be isomorphic if they have Cayley tables which
are identical, except for relabelling of the elements. For example G = {1, i, −1, −i}
is a group with respect to multiplication of complex numbers. It is isomorphic to Z4 ,
because its Cayley table
× 1
1 1
i
i
-1 -1
-i -i
i
i
-1
-i
1
-1 -i
-1 -i
-i 1
1 i
i -1
is identical to that of Z4 , if we relabel elements of Z4 by the rule 0 7→ 1, 1 7→ i,
2 7→ −1, 3 7→ −i (in other words, k 7→ ik , k = 0, 1, 2, 3).
Question Is the group G = {e, a, b, c} in the second example above isomporphic to
Z4 ?
If G is a group containing only two elements, then G is isomorphic to Z2 . To see
this, note that one of the elements of G is the identity e. Let g be the other element
of G. The e ∗ g = g = g ∗ e, where ∗ is the binary operation in G. What is g −1 ? Since
g −1 ∗ g = e 6= g = e ∗ g, g −1 6= e, so g −1 = g. Hence g ∗ g = e, and this determines
the Cayley table of G as
∗
e
g
e g
e g
g e
Clearly this is the same as that of Z2 , using the relabelling 0 7→ e, 1 7→ g.
Exercise Show that any group containing exactly three elements is isomorphic to Z3 .
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Calculations with permutations
Symmetric groups are common examples of finite groups. They are not in general
abelian, as we shall see below. The symmetric group Sn contains n! elements.
A
1 2 ... n
common way to denote permutations is in the form σ =
, where
a
1 a2 . . . an
1 2 ... n
1 2 ... n
ak = σ(k). If τ =
, then σ ◦ τ =
, where
b1 b2 . . . bn
c 1 c 2 . . . cn
ck = abk .
Example
1 2 3
1 3 2
1 2 3
1 2 3
◦
=
.
3 2 1
2 3 1
Example
1 2 3
1 2 3
The group S3 has precisely six elements: ι =
, α =
,
1 3 2
1 2 3 1 2 3
1 2 3
1 2 3
1 2 3
β=
,γ=
,σ=
and τ =
.
3 2 1
2 1 3
2 3 1
3 1 2
Its Cayley table is
◦
ι
σ
τ
α
β
γ
ι
ι
σ
τ
α
β
γ
σ
σ
τ
ι
β
γ
α
τ
τ
ι
σ
γ
α
β
α
α
γ
β
ι
τ
σ
β
β
α
γ
σ
ι
τ
γ
γ
β
α
τ
σ
ι
Note for example that α ◦ β 6= β ◦ α, so that S3 is not abelian.
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Subgroups
A subgroup of a group G is a subset H ⊆ G that is also a group with respect to the
same binary operation as G. Examples include Z as a subgroup of R (with respect
to addition), R∗ and S 1 as subgroups of C∗ with respect to multiplication.
It is important to recognise when a subset of a group G is actually a subgroup of
G. The following result gives a useful criterion.
Theorem (The subgroup test) Let G be a group with respect to a binary operation
∗, and let H be a subset of G. Then H is a subgroup of G if and only if the following
three conditions are satisfied:
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1. Closure: x ∗ y ∈ H ∀x, y ∈ H.
2. Identity: eG ∈ H, where eG is the identity element of G.
3. Inverse: x−1 ∈ H ∀x ∈ H, where x−1 is the inverse of x in G (with respect to
∗).
Examples
1. Let n > 0 be an integer and let nZ = {nx : x ∈ Z}, the set of integers divisible
by n. Then nZ is a subgroup of Z with respect to addition. For the closure
property, note that nx + ny = n(x + y). The identity element is 0 = 0n ∈ Z.
The inverse in Z of nx ∈ nZ is −nx = n(−x) ∈ nZ.
2. Let G be the group of n × n invertible matrices with respect to matrix multiplication, and let H be the set of matrices A ∈ G such that det(A) = 1. Then H is
a subgroup of G. The closure property follows since det(AB) = det(A) det(B);
In ∈ H since det(In ) = 1; and finally if det(A) = 1 then det(A−1 ) = 1.
3. Let G = Sn and let H = {σ ∈ G : σ(1) = 1}. Then H is a subgroup of
G. To check the closure property, if σ, τ ∈ H then σ(1) = τ (1) = 1, so
(σ ◦ τ )(1) = σ(τ (1)) = σ(1) = 1, and so σ ◦ τ ∈ H. Clearly the identity map
sends 1 to 1, so belongs to H. Finally, if σ ∈ H then σ(1) = 1, so σ −1 (1) = 1
and σ −1 ∈ H.
Definition The order of a group G is the number of elements in G (finite or infinite).
It is denoted |G|. Note that |G| ≥ 1, since every group contains at least one element,
namely the identity. The order of an element g ∈ G is the least positive integer k
such that g k is equal to the identity element of G, where g k denotes g ∗ g ∗ . . . ∗ g (k
times). If no such positive integer k exists, then g has infinite order.
Lemma Let G be a group and g ∈ G. Then the set hgi = {g k : k ∈ Z} is a subgroup
of G, and its order is equal to that of g.
(Here g 0 = eG , g −1 is the inverse of g in G, and if k > 0 then g −k denotes (g −1 )k ,
which is the inverse of g k .)
Theorem (Lagrange’s Theorem) Let G be a finite group and H a subgroup of G.
Then the order of H divides that of G.
Corollary Let G be a group and g ∈ G. Then the order of g divides that of G.
Hence, for example, the group S5 contains no element of order 7, since |S5 | = 5! =
60. (In fact, S5 contains elements of orders 1, 2, 3, 4, 5 and 6, but no elements of order
greater than 6.)
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Definition Let H be a subgroup of G, and g ∈ G. Then the left coset of H in G
represented by g is the subset gH = {gh, h ∈ H} of G, and the right coset of H in
G represented by g is the subset Hg = {hg, h ∈ H} of G.
Exercise Show that:
1. the left cosets form a partition of G;
2. the map h 7→ gh is a bijection from H to gH.
Hence prove Lagrange’s Theorem.
Definition A subgroup N ⊂ G is normal if each left coset is also a right coset. (That
is gN = N g for all g ∈ G.) Equivalently, gng −1 ∈ N for all g ∈ G and all n ∈ N .
Example Let G = S5 , H = {σ ∈ S5 , σ(1) = 1}. Then H is a subgroup of G. The
left coset gH is {σ ∈ S5 , σ(1) = g(1)}, while the right coset Hg is {σ ∈ S5 , σ −1 (1) =
g −1 (1)}. If g(1) 6= 1, then these two subsets are different, so H is not normal.
Example Let G = S5 and let A5 be the set of all even permutations in S5 . Then the
left coset gA5 consists of all even permutations (if g is even), or all odd permutaitons
(if g is odd). In other words there are precisely two left cosets A5 and S5 \ A5 .
Similarly there are two right cosets A5 and S5 \ A5 , so right and left cosets are the
same thing, and A5 is normal.
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Homomorphisms
Definition Let (G, ∗) and (H, †) be groups. A map f : G → H is a homomorphism
if
f (x ∗ y) = f (x)†f (y)
∀ x, y ∈ G.
Example The exponential map from (R, +) to (R+ , ×) is a homomorphism (since
exp(x + y) = exp(x) exp(y)).
Lemma Let f : G → H be a homomorphism. Then the image of f ,
Im(f ) := {f (x), x ∈ G} ⊂ H,
is a subgroup of H, and the kernel of f ,
Ker(f ) := {x ∈ G, f (x) = eH } ⊂ G,
is a normal subgroup of G.
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Clearly, a homomorphism f : G → H is surjective iff Im(f ) = H. Less obviously (but easily checked), it is injective iff Ker(f ) = {eG }, the trivial subgroup of
G. A homomorphism that is both injective and surjective (ie bijective) is called an
isomorphism, and two groups G, H are isomorphic (denoted G ∼
= H) if there is an
isomorphism G → H.
(Exercise: understand why this is the same thing as the less formal definition of isomorphic groups given earlier.)
(Another exercise: check that ∼
= is an equivalence relation between groups.)
Example The exponential map from (R, +) to (R+ , ×) is an isomorphism. It is a
homomorphism, and bijective (with inverse ln : R+ → R). Hence the groups (R, +)
and (R+ , ×) are isomorphic.
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Quotient Groups
Let G be a group, and N a normal subgroup. The quotient group (or factor group)
G
of G by N , denoted G/N or N
, is defined to be the set of left cosets gN for all
g ∈ G. (Since N is normal, this is the same as the set of right cosets N g.) The binary
operation on G/N is defined by
(xN )(yN ) := (xy)N
∀ x, y ∈ G.
Of course, one needs to check some things - firstly that the definition does not depend
on the choices x, y of representatives of the two cosets, and then that the resulting
binary operation on G/N satisfies all the axioms for a group. (Exercise: check these
things.)
Example The set 2Z of even integers is a normal subgroup of the group (Z, +). The
quotient group Z/2Z has two elements: 0 + 2Z = 2Z (the set of all even integers and
1 + 2Z (the set of all odd integers). The Cayley table of this group is:
+
0 + 2Z 1 + 2Z
0 + 2Z 0 + 2Z 1 + 2Z
1 + 2Z 1 + 2Z 0 + 2Z
Note that this is just like addition modulo 2. In fact Z/2Z ∼
= Z2 . This is a special
case of an important theorem.
First Isomorphism Theorem Let f : G → H be a homomorphism. Then
G ∼
= Im(f ).
Ker(f )
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