Group in Algebra

Group in Algebra

Let G be a non-empty set and * (multiplication) be a binary operation defined on G. Then algebraic structure (G, *) is called group if it satisfies the following axioms.

1- Closure property

If a∈G, and b∈G then  a*b∈G

it is called closure property. of G

2- Associativity

If a∈G,  b∈G , and c∈G

then

a*(b*c)= (a*b)*c

If G satisfies the the above property, it is called associative.

3-Existence of identity

If a∈G and  e∈G then

ae=ea= a

where e is identity element

4-Existence of inverse

For each a∈G there exists   b∈G

such that

ab=ba=e

Where e is identity

b=a-1

Let us take G as Q then

1- Closure property

If 3∈Q, and 5∈Q then  3*5=15∈Q

Q satisfies closure property.

2- Associativity

If 2∈Q,  3∈Q , and 5∈Q

then

2*(3*5)= (2*3)*5

If Q satisfies the the above property, it is  associative.

3-Existence of identity

If 5∈Q and  1∈Q then

5*1=1*5= 3

where 1 is identity element

4-Existence of inverse

For each 5∈Q there exists   1/5∈Q

such that

5 * 1/5=1/5 * 5=1

Where 1 is identity


PDF file Group in Algebra


Video

 

Leave a Comment

Your email address will not be published. Required fields are marked *

©Postnetwork-All rights reserved.