A subset of GLn consisting of all linear transformations A such that det A = 1is a subgroup of GLn called the special linear group of dimension n and denoted by SLn.. The set of proper o
Trang 1THEOREM The homomorphic image f (G) is a group The image f (e) of the identity element eG is the identity element of the group f (G) Mutually inverse elements of G correspond to mutually inverse images in f (G).
Two groups G1and G2 are said to be isomorphic if there exists a one-to-one mapping
f of G1 onto G2such that f (ab) = f (a)f (b) for all a, bG1 Such a mapping is called an
isomorphism or isomorphic mapping of the group G1onto the group G2
THEOREM Any isomorphism of groups is invertible, and the inverse mapping is also
an isomorphism
An isomorphic mapping of a group G onto itself is called an automorphism of G If
f1: G → G and f2 : G → G are two automorphisms of a group G, one can define another automorphism f1 ◦ f2 : G → G by letting (f1 ◦ f2 )(g) = f1(f2(g)) for all g G This automorphism is called the composition of f1and f2, and with this composition law, the set
of all automorphisms of G becomes a group called the automorphism group of G.
5.8.1-4 Subgroups Cosets Normal subgroups
Let G be a group A subset G1 of the group G is called a subgroup if the following
conditions hold:
1 For any a and b belonging to G1, the product ab belongs to G1
2 For any a belonging to G1, its inverse a–1belongs to G1
These conditions ensure that any subgroup of a group is itself a group
Example 5 The identity element of a group is a subgroup The subset of all even numbers is a subgroup
of the additive group of all integers.
The product of two subsets H1 and H2 of a group G is a set H3 that consists of all
elements of the form h1h2, where h1 H1, h2 H2 In this case, one writes H3= H1H2.
Let H be a subgroup of a group G and a some fixed element of G The set aH is called
a left coset, and the set Ha is called a right coset of the subgroup H in G.
Properties of left cosets (right cosets have similar properties):
1 If aH, then aH ≡H.
2 Cosets aH and bH coincide if a–1bH.
3 Two cosets of the same subgroup H either coincide or have no common elements.
4 If aH is a coset, then aaH
A subgroup H of a group G is called a normal subgroup of G if H = a– 1Hafor any
aG This is equivalent to the condition that aH = Ha for any aG, i.e., every right
coset is a left coset
5.8.1-5 Factor groups
Let H be a normal subgroup of a group G Then the product of two cosets aH and bH (as subsets of G) is the coset abH Consider the set Q whose elements are cosets of the subgroup H in G, and define the product of the elements of Q as the product of cosets Endowed with this product, Q becomes a group, denoted by Q = G/H and called the
quotient group of G with respect to the normal subgroup H.
The mapping f : G → G/H that maps each aG to the corresponding coset aH is a homomorphism of G onto G/H.
If f : G → G is a homomorphism of groups, the set of all elements of G mapped into the identity element of G is called the kernel of f and is denoted by ker f ={g G : f (g) = f (e)}.
Trang 2THEOREM1 If f is a homomorphism of a group G onto a group G and H is the set
of all elements of G that are mapped to f (e) (e is the identity element of G), then H is a normal subgroup in G.
THEOREM2 (ON GROUP HOMOMORPHISMS) If f is a homomorphism of a group G onto
a group G and H is the normal subgroup of G consisting of the elements mapped to the identity element of G, then the group G and the quotient group G/H are isomorphic Thus, given a homomorphism f of a group G onto a group 2 G, the kernel H of the homomorphism is a normal subgroup of G, and conversely any normal subgroup H of G is the kernel of the homomorphism of G onto the quotient group G/H.
Remark. Given a homomorphism of a group G onto a set G, all elements of the group G are divided into mutually disjoint classes, each class containing all elements of G that are mapped into the same element of G.
Example 6 LetRn be the n-dimensional linear coordinate space, which is an abelian group with respect
to addition of its elements This space is the direct product of one-dimensional spaces:
Rn= R1( 1 ) ⊗· · ·⊗ R1(n) Since R 1
(n) is an abelian subgroup, the set R 1
(n) is a normal subgroup of the group Rn The coset corresponding
to an element a Rn is the straight line passing through a in the direction parallel to the straight lineR 1
(n) , and the quotient group Rn R 1
(n)is isomorphic to the (n –1 )-dimensional space Rn–1 :
(n) = R 1
( 1 ) ⊗· · ·⊗ R 1
(n– 1 )
5.8.2 Transformation Groups
5.8.2-1 Group of linear transformations Its subgroups
Let V be a real finite-dimensional linear space and let A : V → V be a nondegenerate linear
operator This operator can be regarded as a nondegenerate linear transformation of the
space V , since A maps different elements of V into different elements, and for any yV
there is a unique xV such that Ax = y.
The set of all nondegenerate linear transformations A of the n-dimensional real linear
space V is denoted by GL(n).
The product AB of linear transformations A and B in GL(n) is defined by the relation
(AB)x = A(Bx) for all xV
This product is a composition law on GL(n).
THEOREM The set GL(n) of nondegenerate linear transformations of an n-dimensional real linear space V with the above product is a group.
The group GL(n) is called the general linear group of dimension n.
A subset of GL(n) consisting of all linear transformations A such that det A = 1is a
subgroup of GL(n) called the special linear group of dimension n and denoted by SL(n).
A sequence{Ak}of elements of GL(n) is said to be convergent to an element AGL(n)
as k → ∞ if the sequence{Akx}converges to Ax for any xV
Types of subgroups of GL(n):
1 Finite subgroups are subgroups with finitely many elements.
2 Discrete subgroups are subgroups with countably many elements.
3 Continuous subgroups are subgroups with uncountably many elements.
Trang 3Example 1 The subgroup of reflections with respect to the origin is finite and consists of two elements:
the identity transformation and the reflection x→ –x.
The subgroup of rotations of a plane with respect to the origin by the angles kϕ (k =0 , 1 , 2, and ϕ
is a fixed angle incommensurable with π) is a discrete subgroup.
The subgroups of all rotations of a three-dimensional space about a fixed axis are a continuous subgroup.
A continuous subgroup of GL(n) is said to be compact if from any infinite sequence of
its elements one can extract a subsequence convergent to some element of the subgroup
5.8.2-2 Group of orthogonal transformations Its subgroups
Consider the set O(n) that consists of all orthogonal transformations P of the n-dimensional Euclidean space V , i.e., P TP = PPT = I (see Paragraph 5.2.3-3 and Section 5.4) This set
is a subgroup of GL(n) called the orthogonal group of dimension n.
All orthogonal transformations are divided into two classes:
1 Proper orthogonal transformations, for which det P = +1.
2 Improper orthogonal transformations, for which det P = –1.
The set of proper orthogonal transformations forms a group called the special orthogonal
group of dimension n and denoted by SO(n).
In the two-dimensional orthogonal group O(2) there is a subgroup of rotations by the angles kϕ, where k =0, 1, 2, and ϕ is fixed If ak is its element corresponding to k and a = a1, then the element a k (k >0) has the form
a k = a⋅a⋅ .⋅a
k times
= a k (k =1, 2, 3, )
Denoting by a–1 the inverse of a = a1, and the identity element by a0, we see that each element of this group has the form
a k = a k (k =0, 1, 2, )
Groups whose elements admit such a representation in terms of a single element are said to
be cyclic Such groups are discrete.
There are two cyclic groups of rotations (p and q are coprime numbers):
1 If ϕ≠ 2πp/q(i.e., the angle ϕ is incommensurable with π), then all elements are distinct.
2 If ϕ =2πp/q, then ak+q = a k (a q = a0) Such groups are called cyclic groups of order q Consider groups of mirror symmetry Each of them consists of two elements: the
identity element and a reflection with respect to the origin
Let{I, P} be a subgroup of O(3) consisting of the identity I and the reflection P of
the three-dimensional space with respect to the origin, Px = –x This is an improper
subgroup It is isomorphic to the group Z2 of residues modulo 2 The subgroup {I, P}
is a normal subgroup in O(3), and the subgroup SO(3) (consisting of proper orthogonal transformations) is isomorphic to the quotient group O(3)/{I, P}
5.8.2-3 Unitary groups
By analogy with Paragraph 5.8.2-2, one can consider groups of linear transformations of a complex linear space
In the general linear group of transformations of a unitary space, one considers unitary
groups U (n), which are analogues of orthogonal groups In the group U (n) of unitary
transformations, one considers the subgroup SU (n) that consists of unitary transformations
whose determinant is equal to1
Trang 45.8.3 Group Representations
5.8.3-1 Linear representations of groups Terminology
A linear representation of a group G in the finite-dimensional Euclidean space V n is a
homomorphism of G to the group of nondegenerate linear transformations of V n; in other
words, a linear representation of G is a mapping D that associates each element aGwith
a nondegenerate linear transformation D(a) of the space V n , so that for any a1and a2in G,
we have D(a1a2) = D(a1)D(a2).
Thus, for any gG, its image D(g) is an element of the group GL(n), and the set D(G) consisting of all transformations D(g), g G, is a subgroup of GL(n) isomorphic to the quotient group G/ker D, where ker D is the kernel of the homomorphism D, i.e., the set of all g such that D(g) is the identity element of the group GL(n).
The subgroup D(G) is often also called a representation of the group G.
The space V n is called the representation space; n is called the dimension of the representation; and the basis in V n is called the representation basis.
The trivial representation of a group is its homomorphic mapping onto the identity element of the group GL(n).
A faithful representation of a group G is an isomorphism of G onto a subgroup of GL(n).
5.8.3-2 Matrices of linear representations Equivalent representations
If D(μ) (G) is a representation of a group G, each gGcorresponds to a linear transformation
D(μ) (g), whose matrix in the basis of the representation D(μ) (G) is denoted by [D(μ)
ij (g)].
Two representations D(μ1 (G) and D(μ2 (G) of a group G in the same space E nare said
to be equivalent if there exists a nondegenerate linear transformation C of the space E n such that D(μ1 (g) = C–1D(μ2 (g)C for each gG.
The choice of a basis in the representation space is important, since the matrices corresponding to the group elements may have some standard fairly simple form in that basis, and this allows one to make important conclusions with regards to a given representation
5.8.3-3 Reducible and irreducible representations
A subspace V ofV n is called invariant for a representation D(G) if it is invariant with
respect to each linear operator in D(G).
Suppose that all matrices of some three-dimensional representation D(G) have the form
A1 A2
O A3
, A1≡
a11 a12
a21 a22
, A2≡
a13
a23
, A3≡( a33) , O ≡(0 0)
The product of such matrices has the form (see Paragraph 5.2.1-10)
A
1 A 2
O A
3
A
1 A 2
O A
3
=
A
1A 1 A 2
3A 3
,
and therefore the structure of the matrices is preserved Thus, the matrices A1 form a
two-dimensional representation of the given group G and the matrices A3 form its
one-dimensional representation In such cases, one says that D(G) is a reducible representation.
Trang 5If all matrices of a representation have the form of size n×n
A1 O
O A2
,
then the square matrices A1 and A2 form representations, the sum of their dimensions
being equal to n In this case, the representation is said to be completely reducible The representation induced on an invariant space by a given representation D(G) is called a part
of the representation D(G).
A representation D(G) of a group G is said to be irreducible if it has only two invariant
subspaces, V n and O Otherwise, it is said to be reducible Any representation can be
expressed in terms of irreducible representations
5.8.3-4 Characters
Let D(G) be an n-dimensional representation of a group G, and let [D ij (g)] be the matrix
of the operator corresponding to the element gG The character of an element gGin
the representation D(G) is defined by
χ(g) =
n
i=1
D ii (g) = Tr([D ij (g)]).
Thus, the character of an element does not depend on the representation basis and is, therefore, an invariant quantity
An element bG is said to be conjugate to the element aG if there exists uGsuch that
uau–1 = b.
Properties of conjugate elements:
1 Any element is conjugate to itself
2 If b is conjugate to a, then a is conjugate to b.
3 If b is conjugate to a and c is conjugate to b, then c is conjugate to a.
The characters of all elements belonging to one and the same class of conjugate elements coincide The characters of elements for equivalent representations coincide
5.8.3-5 Examples of group representations
1◦ Let G be a group of symmetry of three-dimensional space consisting of two elements:
the identity transformation I and the reflection P with respect to the origin, G ={I, P} The multiplication of elements of this group is described by the table
I P
I I P
P P I
1 One-dimensional representation of the group G.
In the space E1, we chose a basis e1and consider the matrix A( 1 )of the nondegenerate
transformation A1of this space: A( 1 )= (1) The transformation A1forms a subgroup in the
Trang 6group GL(1) of all linear transformations of E1, and the multiplication in this subgroup is described by the table
A(1)
A(1) A(1)
We obtain a one-dimensional representation D(1)(G) of the group G by letting D(1)(I) = A(1),
D(1 )(P) = A(1) These relations define a homomorphism of the group G to GL(1) and thus
define its representation
2 A two-dimensional representation of the group G.
In E2, we choose a basis e1, e2 and consider the matrices A( 2 ), B( 2 ) of linear
transfor-mations A2, B2of this space: A(2)= 10 01
, B(2)= 01 10
The transformations A2, B2form
a subgroup in the group GL(2) of linear transformations of E2 The multiplication in this subgroup is defined by the table
A(2) B(2)
A(2) A(2) B(2)
B(2) B(2) A(2)
We obtain a two-dimensional representation D(2)(G) of the group G by letting D(2)(I) = A(2),
D(2 )(P) = B( 2 ) These relations define an isomorphism of G onto the subgroup{A( 2 ), B( 2 )}
of GL(2), and therefore define its representation.
3 A three-dimensional representation of the group G.
Consider the linear transformation A(3)of E3defined by the matrix
A(3 )=
(1 0 0
0 1 0
0 0 1
)
This transformation forms a subgroup in GL(3) with the multiplication law A( 3 )A(3)= A(3)
One obtains a three-dimensional representation D(3)(G) of the group G by letting D(3)(I) =
A(3), D(3)(P) = A(3)
4 A four-dimensional representation of the group G.
Consider linear transformations A(4)and B(4)of E4defined by the matrices
A(4 ) =
⎛
⎜
⎝
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
⎞
⎟
⎠ , B( 4 )=
⎛
⎜
⎝
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
⎞
⎟
⎠
The transformations A( 4 )and B( 4 )form a subgroup in GL(4) with the multiplication defined
by a table similar to that in the two-dimensional case One obtains a four-dimensional
representation D( 4 )(G) of the group G by letting D( 4 )(I) = A( 4 ), D( 4 )(P) = A(B).
Remark. The matrices A(4)and B(4)may be written in the form A(4)=
0 A(2)
, B(4)=
0 B(2)
,
and therefore the representation D( 4 )(G) is sometimes denoted by D( 4 )(G) = D( 2 )(G) + D( 2 )(G) =2D(2)(G) In
a similar way, one may use the notation D(3)(G) =3D(1)(G) In this way, one can construct representations of the group G of arbitrary dimension.
2◦ The symmetry group G ={I, P}for the three-dimensional space is a normal subgroup of
the group O(3) The subgroup SO(3) ⊂ O(3) formed by proper orthogonal transformations
is isomorphic to the quotient group O(3)/{I, P}
Trang 7Since any group admits a homomorphic mapping onto its quotient group, there is a
homomorphism of the group O(3) onto SO(3) This homomorphism is defined as follows:
if a is a proper orthogonal transformation in O(3), its image in SO(3) coincides with a; and if a is an improper orthogonal transformation, its image is the proper orthogonal
transformation P a
In this way, one obtains a three-dimensional representation DO(3) of the group of orthogonal transformations O(3) in terms of the group SO(3) of proper orthogonal
trans-formations
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