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DPP-4 Non abelian Groups

non abelian group assignment dpp by dips academy

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Mathematics

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Sikkim Manipal University

Academic year: 2022/2023

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Dr. Ashutosh Sharma (Ph., NET, JRF, GATE, IIT-JAM)&

Dr. Onkar Singh Bhati (Ph., NET, JRF, GATE, SET)

Innovative Institute of Mathematics, S-10, Mahaveer Nagar, Near Jaipur Hospital, Jaipur

Contacts: 9461777799, 9928363694

1. Let P be a prime number. Let G be the group of

all 2  2 matrices over Z pwith determinant 1

under matrix multiplication. Then the order of

G is

(a) ( p − 1 ) p ( p+ 1 ) (b) p 2 ( p − 1 )

(c) p 3 (d)p 2 ( p − 1 +p)

(JAM MA 2013)

2. Let G be a group of order 8 generated by a and

b such that a 4 = b 2 = 1 and ba = a b. 3 The order

of the center of G is

(a) 1 (b) 2

(c) 4 (d) 8

(JAM CA 2006)

3. Consider the dihedral group

 

2 3 2 3

D 4 = e,r,r ,r , f ,rf ,r f ,r f with

r 4 = e =f 2

and rf = fr − 1 .The product r fr 3 − 1 f − 1 r 3 fr

corresponds to

(a) f (b) rf

(c)

r 2 f

(d)

r f 3

4. Let D 4 is dihedral group of order 8. Let us define

 

2 H = x | x  D 4 be subset of D 4 then number

of elements in H is

(a) 2 (b) 4

(c) 6 (d) 8

5. LetH =  x  D : x 8 8 = ethen cardinality of H is

(a) 8 (b) 4

(c) 16 (d) Cannot say

6. LetGL ( 2 , 7 )be general linear group and

A  GL ( 2 , 7 )such that

4 5

6 3

A

 

=  

 

then

A− 1

is

(a)

4 5

6 3

 

 

 

(b)

4 5

3 6

 

 

 

(c)

1 3

5 6

 

 

 

(d)

4 5

6 3

 

 

 

7. A group G is generated by the elements x, y

with the relations ( )

3 2 2

x = y = xy = e order

of G is

(a) 4 (b) 6

(c) 8 (d) 12

(NET DEC 2015)

8. The number of 2  2 matrices overZ 3 (the field

with three elements) with determinant 1 is

(a) 24 (b) 60

(c) 20 (d) 30

(JAM 2010)

9. Let G be the group of all 2  2 matrices

a b

c d

 

 

 

under matrix multiplication, wheread − bc 0

and a b c d, , , are integers modulo 3. The order

of G is

(a) 24 (b) 16

(c) 48 (d) 81

(DU 2014)

10. Let

2 6

3 5

A

 

=  

 

be a matrix over the integers

modulo 11. The inverse of A is

(a)

8 9

10 9

A

 

=  

 

(b)

10 8

9 9

A

 

=  

 

(c)

9 10

9 8

A

 

=  

 

(d)

9 9

10 8

A

 

=  

 

(DU 2014)

11. The order of the group

| 1 and , , , 3

a b

ad bc a b c d

c d

   

   − =  

   

relative to

matrix multiplication is

(a) 18 (b) 20

(c) 24 (d) 22

(DU 2014)

Innovative Institute of Mathematics, Jaipur

Course: IIT-JAM 2022 Module-1: Group Theory

Unit-2: Group DPP-4: Non abelian Groups

Institute for JAM, NET, M. (Ent.), B., I &II Grade

Dr. Ashutosh Sharma (Ph., NET, JRF, GATE, IIT-JAM)&

Dr. Onkar Singh Bhati (Ph., NET, JRF, GATE, SET)

Innovative Institute of Mathematics, S-10, Mahaveer Nagar, Near Jaipur Hospital, Jaipur

Contacts: 9461777799, 9928363694

12. Inverse of the element

2 6

3 5

 

 

 

inGL ( 2 , 11 )

(a)

5 5

3 8

2  

 

 

(b)

1 5 -

8 -3 2

−   



 

(c)

1 5 5

3 8 2

 

 

 

(d)

9 9

10 8

 

 

 

13. What is the number of non-singular 3  3

matrices overF 2 ,the finite field with two

elements?

(a) 168 (b) 384

(c) 2 3 (d) 32

(Net Dec. 2015)

14. Given a group G, its center is

 h  G hg : = gh for all g  G.Let G be the

group

1 0 1 : , ,

0 0 1

a b

G c a b c

   

   

=     

   

  

. Then the center

of G

(a) Is G itself

(b) Is

1 0

0 1 0 : , ,.

0 0 1

x

a b c

   

   

    

   

  

(c) Consists only of the 3  3 identity matrix

(d)

1 0 1 :

0 0 1

x x

   

   

    

   

  

(IISc 2010)

15. Find the inverse in 5 of the following matrix:

1 2 0

0 2 4

0 0 3

 

 

 

........................

(NBHM 2012)

16. LetH =  x 2 :x  D 4 andK =  x  D 4 :x 2 =e

then

(a) H = K (b) H K

(c) K  H (d) H K

ANSWERS

1. A

2. B

3. C

4. A

5. C

6. C

7. B

8. A

9. C

10. D

11. C

12. D

13. A

14. B

15. 1 1 3

0 3 4

0 0 2

 

 

 

16. C,D

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DPP-4 Non abelian Groups

Course: Mathematics

666 Documents

Students shared 666 documents in this course

University: Sikkim Manipal University

Was this document helpful?

Dr. Ashutosh Sharma (Ph.D., NET, JRF, GATE, IIT-JAM)&

Dr. Onkar Singh Bhati (Ph.D., NET, JRF, GATE, SET)

Innovative Institute of Mathematics, S-10, Mahaveer Nagar, Near Jaipur Hospital, Jaipur

Contacts: 9461777799, 9928363694

1. Let P be a prime number. Let G be the group of

all



matrices over

with determinant

under matrix multiplication. Then the order of

G is

(a)

( ) ( )

1 1p p p

−+

(b)

( )

21pp−

(c)

(d)

( )

21p p p

−+

(JAM MA 2013)

2. Let

be a group of order

generated by a and

b such that

1ab

and

ba a b.

The order

of the center of G is

(a)

(b)

(c)

(d)

(JAM CA 2006)

3. Consider the dihedral group

 

2 3 2 3

D e,r,r ,r , f ,rf ,r f ,r f=

with

r e f==

and

rf fr .

−

The product

3 1 1 3

r fr f r

−−

corresponds to

(a)

(b)

(c)

(d)

4. Let

is dihedral group of order

. Let us define

 

H x | x D=

be subset of

then number

of elements in

(a)

(b)

(c)

(d)

5. Let

 

H x D : x e=  =

then cardinality of

(a)

(b)

(c)

(d) Cannot say

6. Let

( )

2GL ,

be general linear group and

( )

2A GL ,



such that

4 5

6 3

A

=



then

A−

(a)

4 5

6 3







(b)

4 5

3 6







(c)

1 3

5 6







(d)

4 5

6 3







7. A group

is generated by the elements

x, y

with the relations

( )

x y xy e.= = =

The order

(a)

(b)

(c)

(d)

(NET DEC 2015)

8. The number of

22

matrices over

(the field

with three elements) with determinant

(a)

(b)

(c)

(d)

(JAM 2010)

9. Let

be the group of all



matrices







under matrix multiplication, where

0ad bc−

and

, , ,a b c d

are integers modulo

The order

(a)

(b)

(c)

(d)

(DU 2014)

10. Let

2 6

3 5

A

=



be a matrix over the integers

modulo

11.

The inverse of

(a)

8 9

10 9

A

=



(b)

10 8

9 9

A

=



(c)

9 10

9 8

A

=



(d)

9 9

10 8

A

=



(DU 2014)

11. The order of the group

| 1 and , , ,

ab ad bc a b c d







− = 











relative to

matrix multiplication is

(a)

(b)

(c)

(d)

(DU 2014)

Innovative Institute of Mathematics, Jaipur

Course: IIT-JAM 2022 Module-1: Group Theory

Unit-2: Group DPP-4: Non abelian Groups

DPP-4 Non abelian Groups

Mathematics

Sikkim Manipal University

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Preview text

Dr. Ashutosh Sharma (Ph., NET, JRF, GATE, IIT-JAM)&

Dr. Onkar Singh Bhati (Ph., NET, JRF, GATE, SET)

Innovative Institute of Mathematics, S-10, Mahaveer Nagar, Near Jaipur Hospital, Jaipur

Contacts: 9461777799, 9928363694

1. Let P be a prime number. Let G be the group of

all 2  2 matrices over Z pwith determinant 1

under matrix multiplication. Then the order of

G is

(a) ( p − 1 ) p ( p+ 1 ) (b) p 2 ( p − 1 )

(c) p 3 (d)p 2 ( p − 1 +p)

(JAM MA 2013)

2. Let G be a group of order 8 generated by a and

b such that a 4 = b 2 = 1 and ba = a b. 3 The order

of the center of G is

(a) 1 (b) 2

(c) 4 (d) 8

(JAM CA 2006)

3. Consider the dihedral group

 

2 3 2 3

D 4 = e,r,r ,r , f ,rf ,r f ,r f with

r 4 = e =f 2

and rf = fr − 1 .The product r fr 3 − 1 f − 1 r 3 fr

corresponds to

(a) f (b) rf

(c)

r 2 f

(d)

r f 3

4. Let D 4 is dihedral group of order 8. Let us define

 

2

H = x | x  D 4 be subset of D 4 then number

of elements in H is

(a) 2 (b) 4

(c) 6 (d) 8

5. LetH =  x  D : x 8 8 = ethen cardinality of H is

(a) 8 (b) 4

(c) 16 (d) Cannot say

6. LetGL ( 2 , 7 )be general linear group and

A  GL ( 2 , 7 )such that

4 5

6 3

A

 

=  

 

then

A− 1

is

(a)

4 5

6 3

 

 

 

(b)

4 5

3 6

 

 

 

(c)

1 3

5 6

 

 

 

(d)

4 5

6 3

 