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5.8. SOME FACTS FROM GROUP THEORY 229

Example 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 tobe 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
T
T
Euclidean space V , i.e., P P = PP = 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 a k is its element corresponding to k
and a = a 1 , then the element a k (k > 0)has the form

a k = a ⋅ a ⋅ ... ⋅ a = a k (k = 1, 2, 3, ... ).

k times
0
Denoting by a –1 the inverse of a = a 1 , and the identity element by a , 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.
0
q
2. If ϕ = 2πp/q,then a k+q = a k (a = a ). 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 Z 2 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)ofunitary
transformations, one considers the subgroup SU(n) that consists of unitary transformations
whose determinant is equal to 1.