NEB Class 12 Questions: NEB 12 Model Questions 2080/2024


NEB Class 12 Routine 2080-2081: Class 12 Routine

Elementary Group Theory - Exercise 3.4 : Class 12 Math

Complete Exercise 3.4 Elementary Group Theory. Exercise 3.4: Elementary Properties of Group Complete Que Ans note, Class 12 Mathematics.
Elementary Group Theory - Exercise 3.4 : Class 12 Math

Chapter 1: Elementary Group Theory.


Exercise: 3.4

 

Complete Exercise of Elementary Group Theory - Exercise 3.3 : Class 12 Mathematics 2080 NEB.

Exercise 3.4 is about: Elementary Proerties of Group


Read: NEB Class 12 Mathematics All Chapter Exercise



1. If a and b are the elements of a group (G, *) such that

a) a * b = b, prove that a = e

Solution:

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.

Solution:

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.

Solution:

(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]

= e.

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.

Solution:

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 xx = x has a unique solution x = e.

Solution:

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.

Solution:

To prove that G is an abelian group, we need to show that for any elements a and b in G, the operation (binary operation in G) is commutative, i.e., ab=ba; .

Given: (ab)2=a2b2 for all a,bG

Proof:

Let a,bG be arbitrary elements of the group.

From the given condition, we have: (ab)2=a2b2

Expanding the left-hand side of the equation: (ab)2=abab=a2b2

Since (ab)2=a2b2, we can cancel a from the left and b from the right side of the equation:

ab=ba

Thus, for any elements a and b in G, we have ab=ba, which means that G is an abelian group.



Elementary Group Theory - Exercise 3.4 : Class 12 Math PDF



Read:

Getting Info...

About the Author

Iswori Rimal is the author of iswori.com.np, a popular education platform in Nepal. Iswori helps students in their SEE, Class 11 and Class 12 studies with Complete Notes, important questions and other study materials.

Post a Comment

AdBlock Detected!
We have detected that you are using adblocking plugin in your browser.
The revenue we earn by the advertisements is used to manage this website, we request you to whitelist our website in your adblocking plugin.
Site is Blocked
Sorry! This site is not available in your country.