Chapter 1: Elementary Group Theory.
Complete Exercise of Elementary Group Theory - Exercise 3.3 : Class 12 Mathematics 2080 NEB.
Exercise 3.4 is about: Elementary Proerties of Group
1. If a and b are the elements of a group (G, *) such that
a) a * b = b, prove that a = e
a * b = b
or, (a * b) * b-1 = b * b-1
or, a * (b * b-1) = b * b-1
or, a * e = e
or, a = e
b.a * b = e, prove that b = a^-1.
a * b = e
or, a-1 * (a * b) = a-1 * e
or, (a-1 * a) * b = a-1 * e
or, e * b = a-1.
or, b = a-1.
2. If the group ( G, ㅇ) is commutative show that (a ㅇ b)^-1 = a^-1 ㅇ b^-1, for all a, b ∊ G.
(a o b) o (a-1 o b-1) = ((a o b) o a-1) o b-1 [by associative law]
= ((b o a) o a-1) o b-1 [by commutative law]
= (b o (a o a-1)) o b-1 [by associative law]
= (b o e) o b-1 [a o a-1 = e, identity element of G]
= b o b-1 [b o e = b]
Similarly, (a-1 o b-1) o (a o b) = e.
So, a-1 o b-1 is the inverse of a o b.
i.e. (a o b)-1 = a-1 o b-1.
3. Prove that if every element of a group G is its own inverse, then G is abelian.
Given G is a group such that a = a-1 for all a ԑ G. To prove G is abelian, let a,b ԑ G, then a * b ԑ G, where * is the binary operation of G.
Now, (a * b) = (a * b)-1.
= b-1 * a-1 = b * a.
Thus, a * b = b * a for all a,b ԑ G.
So group G is abelian.
4. If ( G, ㅇ) is a group, then the group equation x ㅇ x = x has a unique solution x = e.
x o x = x
Or, x o x = x o e, where e is the identity element of G.
By left cancellation law, we have,
x = e.
Since, identity element of a group is unique, so, x = e is a unique solution of given group equation.
5. If G is a group such that (ab)^2 = a^2.b^2 for all a, b Є G, prove that G is an abelian group.
To prove that G is an abelian group, we need to show that for any elements and in G, the operation (binary operation in G) is commutative, i.e., .
Given: for all
Let be arbitrary elements of the group.
From the given condition, we have:
Expanding the left-hand side of the equation:
Since , we can cancel from the left and from the right side of the equation:
Thus, for any elements and in , we have , which means that is an abelian group.
Elementary Group Theory - Exercise 3.4 : Class 12 Math PDF