Download MTE-06-2008

MTE-06
Assignment Booklet
Bachelor’s Degree Programme
Abstract Algebra
School of Sciences
Indira Gandhi National Open University
New Delhi
2008
Dear Student,
Please read the section on assignments in the Programme Guide for Elective Courses that we sent you
after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for
continuous evaluation. For this you need to answer one tutor-marked assignment for this course,
which is in this booklet.
Instructions for Formating Your Assignments
Before attempting the assignment please read the following instructions carefully.
1) On top of the first page of your answer sheet, please write the details exactly in the following format :
ROLL NO:……………………………..…….
NAME:……………………………..…….
ADDRESS:………………………………..….
……………………………...……..
……………………………...……..
……………………………...……..
COURSE CODE:………………………….
COURSE TITLE:………………………….
ASSIGNMENT NO:………………………
STUDY CENTRE:………………………...
DATE:…………………….….………….
PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND
TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3) Leave a 4 cm. margin on the left, top and bottom of your answer sheet.
4) Your answers should be precise.
5) While solving problems, clearly indicate which part of which question is being solved.
6) The assignment responses are to be submitted to your Study Centre Coordinator by November, 2008,
not later.
Please retain a copy of your answer sheets.
Wish you luck!
2
ASSIGNMENT
(To be done after studying the course material.)
Course Code : MTE-06
Assignment Code : MTE-06/TMA/2008
Total Marks : 100
1. a)
Give an example, with justification, of a relation which is symmetric as well as antisymmetric.
(Note: R is anti-symmetric if (a, b)  R  (b, a)  R.)
(2)
b)
Give an example of three subsets A, B and C, of a universal set U, for which
(2)
A  B  A  C,but B  C.
c)
Apply the principle of induction to show that 21 (4n 1  52n 1 )  n 1.
d)
Let p, q and r be non-zero integers. Prove or disprove the statement, ‘if p q and r p, then r q
(3)
 q   q ’
and     .
p r 
(3)


 a b 

2. Let G =  
a, b,c,dR,ad  bc  0  . Define the product of two elements in G as the usual

c
d





matrix product.
i)
Show that G is a group.
ii)
Find Z(G) = z  G z   z  G.
iii)
Show that K = AG AA t  I  G,and K  G.
iv)
Find a surjective group homomorphism f : G  H, where f (g)  0 for some g  G and H is
an abelian group.
(10)
3. a)


Let G be a finite group and   Aut (G) such that  (x)  x  x  e.


i)
Show that G = g 1  (g) g G .
ii)
If 2  I, show that  (g)  g 1  g G and G is abelian of odd order.
(7)
b)
Find a group of order 120 that contains subgroups isomorphic to each group of order  6.(4)
c)
5
Prove that C
d)
3;
C
C2 , using the Fundamental theorem of homomorphism.
(4)
Let G  {e} be a finite group such that G  G has precisely four normal subgroups. Show
that
i)
G is simple (i.e., it has no proper non-trivial normal subgroups), and
3
ii)
4. a)
G is non-abelian.
(5)
Let R be a subring of a commutative ring S, such that S:R  n. Let m N such that
(m, n) = 1. Prove that R
mR
; S
mS
(6)
.
b)
Check whether the set of continuous functions from R to R is a ring with respect to addition
and multiplication defined by (f + g) (x) = f (x) + g (x) and (f g) (x) = f(g(x))  x  R. (1)
c)
i)
Give an example of a ring R and ideals I and J in R such that IJ  I  J.
ii)
Let I, J be ideals of a ring R such that I + J = R. Prove that I  J = IJ and if IJ = 0,
then R ; R  R .
(8)
I
J
5. a)
b)
  
Find all commutative rings R with unity that have a unique maximal ideal and whose group
of units is trivial.
(3)
Let F be a field, and I  F [x] be the ideal generated by x4 + 2x – 2. For which of the
following choices of F will F[x] be a field, and why?
I
i)
R,
ii)
C,
iii) Q,
iv)
Z2
(4)
c)
Let I = x  7, 15 in Z [x]. Give two distinct non-trivial elements of R =
d)
What is the quotient field of
Zx
.
I
Further, if  : Z [x]  Z [x] :  (f(x)) = f (x + 7), show that  is a ring automorphism and
 (I) = x, 15 . Hence prove that R ; Z15.
(6)
Qx
? Give reasons for your answer. Also check
x 5
whether the characteristic of this field is 5.
(3)
e)
Let R be a ring with at least 2 elements. Suppose that for each non-zero a in R, there is a
unique b  R such that aba = a. Prove that R is a division ring.
(4)
f)
Show that 10 has two distinct factorizations into irreducibles in Z  6  . Hence decide


whether Z  6  is a Euclidean domain.
(5)


6. Which of the following statements are true? Give reasons for your answers.
i)
Every finite abelian group is cyclic.
ii)
{A, , 3, 3, Sh. Manmohan Singh} is a set.
iii)
If R is an integral domain, so is R  R.
iv)
Sn + 1 ; Sn + S1
v)
Char Z[x]
f (x)
 degf (x)
4
vi)
0, 6 is a maximal ideal of Z
vii)
If f (x)  R[x], where R is a ring, then the number of roots of f(x) in R can be more than
deg f.
12.
viii) If D is a UFD, then it is a Euclidean domain.
ix)
 1 2 3 4  2 4 5 6 7 


 is an even permutation.
 3 1 4 2  4 7 6 2 5 
x)
If R [x] is an integral domain, so is R.
(20)
5