UNIT -2
ALGEBRAIC STRUCTURES
Unit-02/Lecture-01
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL, 88-91}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Define group? Distinguish between abelian and non abelian group Explain with the help of example? |
June 2013 |
7 |
|
Q.2 |
Define the following:- (i) Semi group {II} Monoids {iii) Sub Group |
Dec 2013 |
7 |
PROBLEMS ON GROUPS
UNIT- 02/ LECTURE -02
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,91-94}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Show that the set of all positive rational numbers forms an abelian group under composition defined be a*b= ab/2 |
June 2009, June 2011 |
7 |
PROPERTIES OF GROUPS
UNIT- 02/ LECTURE -03
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,94-108}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Prove that inverse of every element of group is unique? |
Dec 2008 |
4 |
|
Q. 2 |
Prove that the left identity is also the right identity i.e if e is the left identity then a o e = a for all a€G |
Dec 2008 |
4 |
PROBLEMS ON GROUPS
UNIT- 02/ LECTURE -04
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,94-108}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Show that the set Q of all rational numbers other than 1 forms an infinite abelian group with operation ‘*’ defined by the rule a * b = a+b-ab |
Dec 2013 |
7 |
|
Q . 2 |
Show that the set of cube roots of unity is an abelian group wrt multiplication |
Dec 2008, June 2014 |
7,2 |
|
Q. 3 |
Show that the set of fourth roots of unity forms an abelian group wrt multiplication |
Dec 2008 |
7 |
SUB GROUPS, CYCLIC GROUP
UNIT-02/ LECTURE -05
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,197-198}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Define the Subgroup |
Dec 2013 |
2 |
|
Q . 2 |
Show that G= {1, -1, i, -i } is a cyclic group under multiplication where i is the imaginary quality such that i2 =-1 |
Dec 2008 |
7 |
|
Q.3 |
Prove that every cyclic group is Abelian? |
June 2014 |
2 |
COSETS, NORMAL SUBGROUPS
UNIT- 02/ LECTURE -06
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,197-198}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
What is a Normal Subgroup? Explain with suitable example |
June 2013, Dec 2013 |
7 2 |
|
Q . 2 |
Let (H, .) be a subgroup of (G,.) .Let N = { x | x € G, xHx-1 = H } Show that (N, .) is a subgroup of G. |
Dec 2012 June 2009 |
7 10 |
|
Q. 3 |
Prove that the intersection of any two normal subgroups of a group is a normal sub group. |
Dec 2013 June 2011, June 2014 |
7 7 3 |
|
Q.4 |
Prove that the order of each subgroup of a finite group is divisor of the order of group |
June 2014 |
7 |
PROBLEMS ON NORMAL SUBGROUPS, FACTOR GROUP
UNIT -02/ LECTURE- 07
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,160-165}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Let H be a subgroup of G then show that xH = Hx forall x € G, x-1H x Ϲ H forall x € G |
Dec 2011 |
7
|
ISOMORPHISM OF GROUPS
UNIT- 02/ LECTURE- 08
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,210-215}
PROBLEMS ON RINGS
UNIT-02/ LECTURE-09
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,227-234}
PROBLEMS ON RINGS, FIELD
UNIT- 02/ LECTURE- 10
REFERENCES {DISCRETE STRUCTURES BY DR.D.C AGARWAL,245-248}
|
S.NO |
RGPV QUESTIONS |
Year |
Marks |
|
Q.1 |
Prove that a ring R is commutative if and only if (a + b)2= a2 + 2ab + b2 forall a,b € R |
June 2007 |
7
|
|
Q. 2 |
If R is a ring such that a2 = a forall a € R prove that (i) a + a =0 forall a € R i.e each element of R is its own additive inverse (ii) a+b =0 => a=b, b € R (iii) R is commutative ring |
Dec 2011 June 2013 |
7 |
|
Q.3 |
Define Ring. If a, b, c are arbitrary elements of a ring R, prove that (i) a0= 0a (ii) a(-b)=-(ab)=(-a)b |
June 2014 |
7 |