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fo’k; % xf.kr
Subject : Mathematics
dkslZ “kh’kZd % Abstract Algebra
Course Title : Abstract Algebra
vf/kU;kl (Assignment) 2012-2013
Lukrd dyk dk;ZØe
Under Graduate Art Programme
Section - A
[k.M & d
fo’k; dksM% ;w0th0,e0,e0
Subject Code : UGMM
dkslZ dksM % ;w0th0,e0,e0-06
Course Code : UGMM-06
vf/kdre vad % 30
Maximum Marks:30
vf/kdre vad % 18
Max. Marks: 18
uksV % nhkZ mRrjh; iz”uA vius iz”uksa ds mRrj 800 ls 1000 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA
Note : Long Answer Questions. Answer should be given in 800 to 1000 words. Answer all
questions.
1 Define a semi-group and a group. Prove that a semi-group having finite number of
elements in which cancellation laws hold is a group. Does this result hold for an
infinite semi-group 6
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fu;e lEHko gS] og ,d lewg gSA D;k ;g ckr vifjfer vo;o okys lewg esa Hkh ykxw gksrk gS\
2 Show that the set Z m of residue classes modulo the positive integer m is a
commutative ring with identity with respect to addition and multiplication of residue
classes. 6
fn[kk;s fd leqPp; Z m ¼jsftM~;w oxZ mod m½ ,d Øefofue; oy; identity vo;o ds lkFk gS]
jsftM~;w oxZ ds ;ksx ,oa xq.ku ds lkis{k esaA
3 Deifne a subgroup of a group and find a necessary and sufficient condition for a nonempty
subset of a group to be a subgroup of the group.If
homomorphism, prove that the Ker f is a suchgroup of G.
6
'
f : G ® G is a group
fdlh lewg ds milewg dks ifjHkkf’kr djsaA fdlh v”kwU; lewPp;] fdlh lewg dk milewg ds fy,
'
f : G ® G ,d lewg ledkfjrk gS rks fn[kk;sa fd
vko”;d ,oa fuf”pr “krZ Kkr djsaA ;fn
(Ker f ), G dk ,d milewg gSA
Section B
[k.M & [k vf/kdre vad % 12
Max. Marks: 12
uksV % ykq mRrjh; iz”uA vius iz”uksa ds mRrj 200 ls 300 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA
Note : Short Answer Questions. Answer should be given in 200 to 300 words.
Answer all questions.
4 Let A and B be two finite sets having m and n elements respectively. Find the number
of distinct relations that can be defined from A to B. 3
nks ifjfer leqPp; A rFkk B ftuesa m, n vo;o gS ds chp fdrus vyx&vyx lEcU/k gks ldrs gSA
5 In a group with even number of elements, show that there is at least one element
besides the identity, which is its own inverse and there will be odd number of such
elements. 3
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vo;o gSaA fn[kkb;sa fd [kqn izfrykse okys vo;o gSA fn[kkb;sa fd [kqn izfrykse okys vo;oksa dh la[;k
fo’ke gksxhA
6 Let f be a group homomorphism from the additive group of integers to the
multiplicaitve group of non-zero rational numbers. Find the following 3
(i) f(0)
(ii) f(-2) if f(2)=1
(iii) f(3) if f(1)= -1
;fn f : ( z,
+ ) ® ( Q -{0};
) ,d ledkfjrk gS rks fuEu dks Kkr djsaA
(i) f(0)
(ii) f(-2) ;fn f(2)=1
(iii) f(3) ;fn if f(1)= -1
7 Define a cyclic group and give an example of a non-cyclic group whose all proper
subgroups are cyclic. 3
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pØh; gksA