Find the zeros and singularities of the function  in . Also find the residue at the poles.

. Which of the following statements are true, and which are false? Give reasons for your answers.  
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and  is of degree , then f(x) has exactly n roots in R.

 Expand  in a Laurent series valid for 

i) 0 < |z - 1| < 2 and  ii) 0 < |z - 3| < 2.

 Is , for any two ideals I and J of a ring R? Give reasons for your answer.  Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and  on S by:


.
Show that  is a ring isomorphic to R.  
 

 Is , for any two ideals I and J of a ring R? Give reasons for your answer. 

For an ideal I of a commutative ring R, define
. Show that
i)  is an ideal of R.
ii) .
iii)  in some cases. 

Which of the following statements are true, and which are false? Give reasons for your answers. 
i) For any ring R and .
ii) Every ring has at least two elements.
iii) If R is a ring with identity and I is an ideal of R, then the identity of R/I is the same as the identity of R.
iv) If  is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +).
v) If R is a ring, then any ring homomorphism from  into R is surjective.

Find the radius of convergence of the following series.
i)   ii) 

 Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H' and K' be normal subgroups of H and K, respectively. Then
i) 
ii) 

iii)   
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.

PART-B (MM: 30 Marks)
(Based on Block 3.)

 Give the smallest  for which A_n is non-abelian. Justify your answer. 

 Evaluate , where c is the eight like figure shown in Fig. 1.

 List two distinct cosets of  in D₁₀, where r is a reflection in D₁₀. 

Find the maximum modulus of  on the closed circular region defined by .

 Let  be a fixed odd permutation in S₁₀. Show that every odd permutation in S₁₀ is a product of  and some permutation in A₁₀.

Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.

 Evaluate  where c is the circle .

Prove that a cyclic group with only one generator can have at most 2 elements.

Find all the singularities of the function 

 Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that  is not cyclic. 

 Find the constant c such that  can be extended to be analytic at , when  is fixed.