Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
5
Unit-I
Definition
Group
A non empty set of elements G is said to form a group if in G there is defined a binary operation, called the
product, denoted by., such that:
1. a. b∈G ∀a,
b∈ G (closed)
2.
(associative law)
3. ∃ an element such that a.e = e.a = a (the existence of an identity element in G)
4.
such that
a. b = b.
a = e (The existence of an identity element in G)
Example 1:
Let
fgge ()
− 1
e
∀a . ∈
∈. c gb = , F i.e.
∃b . ∈b g
. c∀a, b,
c ∈G
G A= a a aij
Rational numbers Q A
H I G is the set of nonsingular 2×2 matrix over rational numbers Q.
1
∴ ≠ 0a . Rb 11 G 12forms S
a a K J a group under matrix–multiplication. U Infact, we note that
: ∈ ,det( ) ≠0
T
F ∈ ∀ , b ∈G
a = HG a a I
2=
V
a a KJ
11 12
,
21 22
F W
1. Let a = HG a a I b b
b
a a KJ F = HG I
21 22
b b KJ
11 12
11 12
, be two non-singular 2×2 matrices over Q.
21 22
21 22
Now a.b under matrix multiplication is again 2×2 matrix over Q and det (a.b) = (det a) (det b) ≠ 0 , as
det a , det b .
2. We know that matrix multiplication is always associative. Therefore,
b g
b g
a. b . c = a. b. c ∀a, b,
c ∈G
1 0
e I G a I I a a a G
0 1
F 3. ∃ =
H G I K J = ∈ suchthat . = . = ∀ ∈
4. If a ∈ G, say
we get a − 1 1
=
a a − a a
b
11 22 21 12
then
F
g
a
HG
−a
−a
22 12
a
21 11
I
KJ