attachments / l12_alg_eng020114
.docLECTURE 12
Groups
A
non-empty set G
is called a group
if a binary algebraic operation is given in G
(often called multiplication),
i.e. a unique element
is determined for every ordered pair
(a, b) of
elements of G
(their product) and the following conditions hold:
1.
for all a,
b, c
G
(the multiplication is associative).
2.
There is a unit
e
G
such that
for all a
G
(existence of unit).
3.
For any a
G
there is an inverse
element
such that
(existence
of inverse element).
A
group G
is called commutative
or abelian
if
for all a,
b
G.
In an abelian group the binary operation is sometimes called addition
and therefore the product is replaced by the sum and is denoted by a
+ b.
In this case a unit is denoted by zero 0, and an inverse for a
element is called opposite
is denoted by –
a.
Examples
of groups:
1) The set
of real numbers regarding to the ordinary operation of addition forms
a group, and
is the number 0. This group is denoted by
.
2)
The set
of positive real numbers forms a group regarding to the operation of
multiplication, and
is the number 1. It is denoted by
.
3) The set of turns of plane around a fixed point forms a group regarding to the operation of composition.
4) The set of affine transformations of plane forms a group regarding to the operation of composition.
Theorem 1. There is the only unit in a group.
Proof.
Let
be a group. Assume the contrary: there are at least two units
and
of
.
Then
since
is a unit of the group, we have
for all a
G.
Consequently,
.
On other hand, since
is a unit of the group, we also have
.
Thus,
.
Theorem 2. There is the only inverse element for every element of a group.
Proof.
Let
be a group. Assume the contrary: there exists an element
having at least two inverse elements
and
for
,
i.e.
and
.
Then we have:
.
The number of elements of a group G (if it is finite) is called the order of group G is denoted by |G|. In this case G is called finite. If the set G is infinite then the group G is called infinite.
Example 1. Let G be the set consisting of numbers 1 and – 1, and let a binary operation be the multiplication of numbers. Does the set G form a group? If yes, which order does the group have and is this group abelian?
Solution: First of all observe that in result of multiplication of two numbers of the set G we obtain the number also belonging to this set:
i.e. for all a,b of G we have ab G.
The condition 1 holds since the multiplication of numbers is associative. The number 1 is the unit of G. 1 is inverse for 1, and – 1 is inverse for – 1. Thus, all conditions hold, and consequently the set G is a group. Its order is equal to the number of elements of G, i.e. it is equal to 2. Since the multiplication of numbers is commutative then this group is abelian.
Integer
powers (degrees) of an element a
G
are determined recurrently:
for natural n,
for integer negative n.
A group G
is called a cyclic
group with a generating element a
if all elements of the group G
are integer powers of the element a.
The
least natural number n
for which
(if there is) is called the
order (period) of element a.
If
for all n
then a
is called an element of infinite
order.
Example 2. Let Z3 be the set consisting of numbers 0, 1 and 2, and let a binary operation be the addition of numbers by module 3. Does the set Z3 form a group? If yes, is it cyclic? If this set is a cyclic group, determine a generating element.
Solution: The addition by module 3 is determined as follows:
We see that for all a,b of Z3 we have a + b Z3.
The addition by module 3 is associative and commutative. The number 0 is the zero-element of Z3. 1 is inverse for 2, and 2 is inverse for 1. Thus, Z3 is a group. The number 1 is a generating element of Z3, i.e. Z3 is a cyclic group.
A
subset H
of a group G
is called a subgroup
of group G
if H
is a group with respect to the operation given in G.
If
is a subgroup of
,
we denote this by
.
If
and
,
we denote this by
.
For example the set H = {1} from Example 1 is a subgroup of the group G.
Example of subgroups.
1)
The set
of integers with the operation of addition is a subgroup of the group
.
2)
The set
of positive rational numbers with the operation of multiplication is
a subgroup of the group
.
A
subgroup H
of a group G
is called normal
in G
if for any elements h
H, g
G
the element
also belongs to H.
An element of group of kind
is called conjugate
with an element h
by g.
If
is a normal subgroup of
,
we denote this by
.
Example.
is a normal subgroup of
.
Two
groups G1
and G2
(with operations
and
respectively) are called isomorphic
if there such one-to-one correspondence (bijection) mapping
that for any two elements a
and b
of G
the following equality holds:
The
notation of isomorphism of groups:
.
Let
H
be a subgroup of G.
A left
coset of an element g
G on the subgroup H
is called the set
A
right
coset
is determined analogously.
The group G is partitioned into pairwise non-intersecting left (right) cosets on the subgroup H, and the cardinality of any coset is equal to the cardinality of H.
Theorem of Lagrange. The order of a finite group is divided on the order of any its subgroup.
Proof.
Let
be a group, and
is a subgroup of
.
Let
,
i.e.
is the order of
,
and
.
Obviously,
.
Consider all left cosets of
on
:
.
We have:
for all
,
for every
,
and
.
Then obviously
,
i.e.
is
divided on
.
In
general
.
If a subgroup H
is normal in G
then
for all g
G.
And in this case the set G/H
of cosets of group G
on subgroup H
is a group with respect to the operation of multiplication of cosets
defined by the following equality:
This group is called the factor group of group G on normal subgroup H.
Example.
Consider
– the group of integers with the operation of addition. Let
be the set of integers that are multiple to 5, i.e. the following
set:
.
Then obviously
with the operation of addition is a subgroup of
.
Consider
the left cosets on the subgroup
:
where
.
Then
is partitioned into pairwise non-intersecting left cosets
and
.
For
example,
.
Observe that
is a normal subgroup of
,
and the factor group
consists
of five elements
and
with the operation of addition
where
is the addition by module 5. Obviously,
.
Glossary
coset – смежный класс; conjugate – сопряженный
generating element – порождающий элемент; factor group – фактор-группа
power – степень
Exercises for Seminar 12
12.1. Does the following set form a group with respect to the operation of addition:
1) the set of real numbers;
2) the set of all negative real numbers;
3) the set of rational numbers;
4) the set of complex numbers;
5) the set of all pure imaginary complex numbers;
6) the set consisting of one number 0?
12.2. Does the following set form a group with respect to the operation of multiplication:
1) the set of real numbers;
2) the set of all positive real numbers;
3) the set of all non-zero complex numbers;
4) the set of all complex numbers equal to 1 by module;
5) the set of complex roots of the n-th power from 1 (n is a natural number)?
12.3. Prove that for any group the following holds:
1) a unit e is unique;
2) an inverse element a –1 is unique for any element a;
3)
the equality ax
= b
is equivalent to
,
and the equality xa
= b
is equivalent
;
4)
for any elements a
and b
the equality
holds.
12.4. Prove that the group of real numbers with respect to the operation of addition is isomorphic to the group of positive real numbers with respect to the operation of multiplication.
12.5. Prove that the following group is cyclic and find its generating element:
1) the group of all integer numbers with respect to addition;
2) the group Un of complex roots of the n-th power from 1 with respect to multiplication.
12.6. Find (up to isomorphism) the factor group G/H if:
1) G is the group of all complex numbers with the operation of addition, H is the subgroup of all real numbers.
2) G is the group of non-zero complex numbers with the operation of multiplication, H is the subgroup of numbers equal to 1 by module.
3) G = Z is the group of integer numbers with the operation of addition, H = nZ is the subgroup of numbers multiple to a given natural number n.
Exercises for Homework 12
12.7. Let G be the set consisting of four complex numbers: 1, i, – 1, – i, and let a binary operation is the multiplication of complex numbers. Prove that the set G with this operation is an abelian group. Determine the order of the group.
12.8. Does the following set form a group with respect to the operation of addition:
1) the set of all non-negative real numbers;
2) the set of all positive rational numbers;
3) the set of integer numbers;
4) the set of even numbers;
5) the set of odd numbers?
12.9. Does the following set form a group with respect to the operation of multiplication:
1) the set of all negative real numbers;
2) the set of rational numbers;
3) the set of natural numbers;
4) the set of all complex numbers;
5) the set of all non-zero pure imaginary complex numbers?
12.10. Prove that if the square of any element of a group is equal to unit then the group is abelian.
12.11. Prove that the following group is cyclic and find its generating element: the group nZ of integer numbers multiple to a given natural number n with respect to addition.
12.12. Find (up to isomorphism) the factor group G/H if:
1) G is the group of non-zero complex numbers with the operation of multiplication, H is the subgroup of positive real numbers.
2) G is the group of all real numbers with the operation of addition, H is the subgroup of integer numbers.