MATHEMATICS MODEL PAPER THIRD SEMESTER  ABSTRACT ALGEBRA commom for B.A and BSC 2017
https://www.computersprofessor.com/2017/11/mathematicsmodelpaperthirdsemester.html
MATHEMATICS MODEL PAPER
THIRD SEMESTER  ABSTRACT ALGEBRA
commom for B.A and BSC 2017
SECTION
– A


I.

Answer
any five questions. Each question carries five marks.

5
x 5 = 25 M


1.

Show that a group G is abelian if and only
if (ab)^{2} = a^{2}b^{2} " a, b ÎG.


2.

Define order of an element in a group and
prove that in a group G, O(a) = O(a^{–1}) for a Î G.


3.

Prove that intersection of two sub groups
of a group G is a sub group of G.


4

Show that a sub
group H of a group G is normal iff xHx^{–1} = H"x ÎG.


5.

Show that the mapping f: G ® G such that f(a) = a^{–1} " a Î G is an
automorphism of a group G iff G is abelian.


6.

If f is a
homomorphism of a group G in to a group G’ then prove that kernel of f is a
normal sub group of G.


7.

Express the product
(2 5 4) (1 4 3) (2 1) as a product of disjoint cycles and find its inverse.


8.

Prove that an infinite cyclic group has
exactly two generators.


SECTION – B


II.

Answer
the all five questions. Each carries ten marks

5
x 10 = 50 M


9.

a)

Show that a finite semi group satisfying
cancellation jaws is a group.



(OR)



b)

Show that the set G of rational numbers
other than 1 is an abelian group with respect to the operation Å defined by a Å b = a + b – ab "a, b ÎG.


10.

a)

Prove that the necessary and sufficient
condition for a non empty sub set H of a group G to be a sub group is that a,
b Î H Þ ab^{–1}ÎH.



(OR)



b)

State and prove Lagrange’s theorem.


11.

a)

If H is a normal sub group of a group G
then prove that the set g/H of all cosets of H in G is a group with respect to
coset multiplication.



(OR)



b)

Prove that a subgroup H of a group G is a
normal subgroup of G iff the product of two right cosets of H in G is again a
right coset of H in G.


12.

a)

If f is a homomorphism of a group G in to a
group G’ then prove that f is an into isomorphism iff ker f = {e}



(OR)



b)

State and prove fundamental theorem on
homomorphism.


13.

a)

State and prove Cayley’s theorem.



(OR)



b)

Prove that every sub group of a cyclic
group is cyclic.


THE
END