Old Exam Papers June 2012 (Set 2)

prove that G' H if and only if H is a normal
subgroup of G and G
H
derived subgroup of G.
is abelian. G' denoted
(b) Prove that an infinite abelian group does not have
a composition series.
3. (a) Let a and b be two non-zero elements of a
Euclidean ring R such that b is not a unit in R, then
b g bg
prove that d a, b d a .
(b) Let M and M' be two R-modules. If f : M M'
is an R-module homomorphism, then prove that f
is a monomorphism if and only if kernel
fbgl 0 q.
4. (a) Let R be the field of real numbers, then
show that the map 2 3
t: R R defined by
t ba , bgb a b, a b, bg
ba , bg R
2 is a
linear transformation.
(b) Let V be a finite dimensional vector space over a
field F, then prove that for each non-zero vector
v V , there exists a linear functional
f V * such that f vbg 0 .
800 2 MA/M.Sc.-MT-01
n
2 2
v v, ui
i1
10. (a) If W is any subspace of a finite dimensional inner
c h
product space V, then prove that W W
where W denotes orthogonal complement of W.
(b) Let V and V' be inner product spaces. Then prove
that a linear transformation t: V V'
is orthogonal
if and only if :
t bg u u u V
MA/M.Sc.-MT-01 5 800