Download 2. Groups

Introduction to Group Theory (cont.)
1. Product Notation
The product of n elements a1, a2, … an of a set G with multiplication is
inductively defined as follows:
1
a
i 1
1  a1
if n  1;
n
a
i 1
i
n 1
 ( ai )an
if n  1.
i 1
This product is also written as a1·a2· … ·an. We have the following
generalization of the associative law.
Proposition 1: Let n0, n1, … nr be integers such that 0 = n0 < n1 < … < nr = n.
Then
 nj
 n
a
ai .



  k 
j 1  k n j 1 1
i

1

r
This is clear for r = 1. Hence we can assume that it is true for r  1 and prove it
for r factors. The details are left as a homework exercise.
2. Groups
Definition: A set G with a multiplication is called a group if the following
(group) axioms are satisfied:
[G1]
The set G is not empty.
[G2]
If a, b, c  G, then (ab)c = a (bc).
[G3]
There exists in G an element e such that
(1) For any element a in G, ea = a.
(2) For any element a in G there exists an element a’ in G such that
a’a = e.
In view of axiom [G2] and the general associativity law, we can and will write the
product of any finite member of elements of G without inserting parentheses.
Proposition: If G is a group and e the element specified in [G3(1)], then e is
an identity element.
Proposition: If G is a group, a an element of G, then a’ (specified in [G3(2)]
is its inverse element.
Proposition: If G is a group, then each of the equation ax = b and xa = b in
an unknown x has a unique solution.
3. Examples for Groups
The following are groups:
{0,1} with multiplication given by the following table:
· 0 1
0 0 1
1 1 0
{0,1,2} with multiplication given by the following table:
·
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
{0,1,2,3} with multiplication given by the following table
·
0
1
2
3
0
0
1
2
3
1
1
0
3
2
2
2
3
0
1
3
3
2
1
0
{0,1,2,3} with multiplication given by the following table
·
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
4. Subgroups
Definition: Let (G, ·) be a group and (H,) another one. Then (H,) is a
subgroup of (G, ·) if and only if H  G and a,b  H: ab = a·b.
With other words, a subgroup of G is a subset with the same multiplication that is
itself a group.
Proposition: Let H  G be a subgroup. Then
The identity element of H is the identity element of G .
Proposition: A non-empty subset H of G is a subgroup (with the same
multiplication) iff
a, b  H : ab 1  H.