Total No. of Questions [Total No. of Pages 02
M.Sc. (Previous) DEGREE EXAMINATION, MAY- 2018
First Year
MATHEMATICS
Algebra
Time Hours Maximum Marks :70
SECTION A
Answer any five questions.
All questions carry equal marks.
Q1) State and prove Sylow's theorem for abelian groups.
If G is a group then show that the set of automorphisms of G is also a
group.
Q2) Show that every group is isomorphic to a subgroup of for some
appropriate S.
Show that conjugacy is an equivalence relation onG.
Q3) Let G be a group and if G in the internal direct product of N1 N2 … Nn then
show that G is isomorphic to N1× N2× … ×Nn.
Describe all finite abelian groups of order 24 34.
Q4) Show that a finite integral domain is a field.
If R is a commutative ring with unit element and M is an ideal then show
that M is a maximal ideal of R if and only if R/M is a field.
Q5) Show that every integral domain can be imbedded in a field.
Q6) If L is a finite extension of K and if K is a finite extension of then show
that L is a finite extension of F in particular
If is a polynomial in of degree n≥1 and irreducible over F then
show that there is an extension E of F such that n in which has a
root.
Q7) Show that the polynomial has a multiple root if and only if
and f ′ have a nontrivial common root.
If K is finite extension of then show that G is a finite group with its
order satisfie
(DM01)
Q8) Show that a group G is solvable if and only if e for some integer k.
Show that the general polynomial 1 ... n n
P x x a x an for n≥5 is not
solvable by radicals.
Q9) Show that a lattice of invariant subgroups of any group is modular.
If a andb are any two elements of a modular lattice then show that the
intervals I and I are isomorphic.
Q10)Show that if L is a complemented modular lattice that satisfies both chain
conditions, then the element 1 of L is a 1∪b of independent points and
conversely if L is a modular lattice with 0 and 1 such that 1 is a 1∪b of a finite
number of points then L is complemented and satisfies both chain conditions.