M.Phil. DEGREE EXAMINATION, APRIL 2016
Mathematics
RESEARCH METHODOLOGY FOR ALGEBRA
(2015 onwards)
Time 3 Hours Maximum 75 Marks
Answer all questions. 15 75)
All questions carry equal marks.
1. Define tower and refinement. Let G be a finite
group. Prove that an abelian tower of G
admits a cyclic refinement.
Let H be a normal subgroup of a group G .
Prove that G is solvable if and only if H and
H
G are solvable.
Or
If n prove that the alternating group An
is simple.
Prove that a group of order 35 is cyclic.
2. Construct the Grothendieck group of a
commutative monoid. If the cancellation law
holds in M prove that the canonical map
M is injective.
Sub. Code
571101
RW-11072
2
Sp4
If A B is a finite abelian group, prove
that . Deduce that a finite
abelian group is isomorphic to its own dual.
Or
Show that the completion and the inverse
limit
H
lim G are isomorphic under natural
mappings.
Let A and B be two groups whose
set-theonetic intersection is Prove that
there exists a group. A B containing A,B as
subgroups such that A along with
the property Every element of A B has a
unique expression as a product
a1....an with ai A or ai
and such that if ai then and if
ai then A .
3. Prove that every ideal is contained in some
maximal ideal.
State and prove Chinese Remainder theorem.
Or
Prove that every principal entire ring is
factorial.
Describe the construction of .
RW-11072
3
Sp4
4. Prove that a sequence y" is exact
if and only if

is exact for all x .
Prove that HomA
HomA HomA as abelian
groups.
Or
Define abelian category. Give an example.
Let V be a vector space over K and let W be
a subspace. Prove that
W V
dimK V dimKW dimK .
5. Let F be a free module, and M a submodule.
Prove that M is free and its dimension is less
than or equal to the dimension of F .
State and prove Snake lemma.
Or
Prove that the direct limits exist in the
category of modules over a ring.
Assume that An satisfies ML. Given an
exact sequence O O
g
n n
of inverse systems, prove that
O is exact.