Ques 1.

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)

i) where  is the Euler-phi function.

ii) If  and  are groups, and  is a group homomorphism, then 

iii) If G is an abelian group, then G is cyclic.

iv) If G is a group and ,then  

v) Every element of  has order at most  n .

vi) If R is a ring and I is an ideal of R , then I xr = rx ∀ x ∈ I and r ∈ R .

vii) If  is a product of an even number of disjoint cycles, then sign 

viii) If a ring has a unit, then it has only one unit.

ix) The characteristic of a finite field is zero.

x) The set of discontinuous functions from  to  form a ring with respect to pointwise addition and multiplication.

Ques 2.

Define a relation R on  by R =  Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on 

Ques 3.

Consider the set  Define  on by 

i) Check whether  is a group or not.

ii) Prove that x ∗ x ∗ x ∗K∗ x (n times) = (1+ x) −1 ∀ n∈N n and x ∈X .

Ques 4.

Give an example, with justification, of a commutative subgroup of a noncommutative group.

Ques 5.

Check whether or not A  is a subgroup of 

i)                        ii)     

Ques 6.

Let  be a finite abelian group and  Prove that 

Ques 7.

Let G be a group of order n ≥ 2 , with only two subgroups  and itself. Find a minimal generating set for G . Also, find out whether n is a prime or a composite number, or can be either.

Ques 8.

Consider the map  Let  Then B is a group with respect to the composition of functions. Check whether or not  is a normal subgroup of B .

Ques 9.

Explicitly give the elements and structure of the group 

Ques 10.

Let G be a group of order 56 . What are all its Sylow p-subgroups? Show that G is not simple, i.e., G must have a proper normal non-trivial subgroup.

Ques 11.

Find a group G , and a homomorphism « of G , so that  and Ker   Is G abelian? Give reasons for your answer.

Ques 12.

Let G be a group such that G Aut is cyclic. Prove that G is abelian.

Ques 13.

Check whether   is a subring of the ring ) ( M2 Z or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.

Ques 14.

Prove that      as rings.

Ques 15.

Find all the units of 

Ques 16.

Let R be a commutative ring with unity and r ∈ R . Prove that   using the Fundamental Theorem of Homomorphism. Hence show that 

Ques 17.

Let Check whether D is a UFD or not.

Ques 18.

Let 

i) Show that M is an ideal of R .

ii) Show that if a|5 / or b/|5 , then 5|  for ,a . b ∈ 

iii) Hence show that if N is an ideal of R properly containing M , then N = R .

iv) Show that  is a field, and give two distinct non-zero elements of this field.

Ques 19.

Show that there are infinitely many values of α for which  is irreducible in  .