Page 33 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Now (a * a) * b = a * b = e.
Also a * (a * b) = a * e = a. Therefore, by Gl’, a = e.

Now we will use this lemma to prove Theorem 3.
Proof of Theorem 3: G1 holds since G1 and G1' are the same axiom. We will next prove that G3
is true. Let a  G. By G3' 3 b  G such that a * b = e. We will show that

b * a = e. Now,
(b * a) * (b * a) = (b * (a * b)) * a = (b * e) * a = b * a .
Therefore, by Lemma 1, b * a = e. Therefore, G3 is true.
Now we will show that G2 holds. Let a  G. Then by G2', for a  G, a * e = a. Since G3 holds,
 b  G such that a * b = b * a = e. Then
e * a = (a * b) * a = a * (b * a) = a * e = a .
That is, G2 also holds.

Thus, (G, *) satisfies G1, G2 and G3.

Example: Let G = { ±1,  i }, i =  Let the binary operation be multiplication. Show
1.
that (G) is a group.
Solution: The table of the operation is

1 –1 i –i
1 1 –1 i –i
–1 –1 1 –i i
i i –i –1 1
–i –i i 1 –1

This table shows us that a.l = a  a  G. Therefore, 1 is the identity element. It also shows us that
(G) satisfies G3. Therefore, (G) is a group.
Note that G = {1, x, x , x }, where x = i.
3
2
2.2.1 Abelian Group

Definition: If (G, *) is a group, where G is a finite set consisting of n elements, then we say that
(G, *) is a Finite group of order n. If G is an infinite set, then we say that (G,*) is an infinite group.

If * is a commutative binary operation we say that (G, *) is a commutative group, or an abelian
group. Abelian groups are named after the gifted young Norwegian mathematician Niels Henrik
Abel.
Now let us discuss an example of a non-commutative (or non-abelian) group. Before doing this
example recall that an m x n matrix over a Set S is a rectangular arrangement of elements of S in
m rows and n columns.

 a b 
If A =   then ad-bc is called the determinant of A and is written as det
Note  c d
A or |A|

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