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
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