Let ? (?) be a finite extension F of odd degree(greater than 1). Show that ? (?²) = ? (?).

Compute the following:

(i)             (ii)      

Is there a finite group with class equation 1+1+2+2+2+2+2+2?

Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.

Describe the set of primes p for which x² - 11 splits into linear factors over Zp. 

Find the elementary divisors and invariant factors of 

Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that Pis normal in K.

Consider the natural action of GL₂ on M₂, the set of 2 x 2 real matrices, by left multiplication.

(i) Under this action, if det(x) ≠ 0, show that the stabiliser of x ∈ M₂ is {I}, where I is the 2 x 2 identity matrix.

(ii) Suppose that det(x) = 0 in the remaining parts of this exercise. We will show that the stabiliser of x is infinite. If x = 0, the stabiliser of x is GL₂. So suppose x ≠ 0. Let us writeThen,  for non-zero λ ∈ R. Why?

(iii) Let  be a vector that is not a scalar multiple of  . Show that there is a matrix b =  such that b  = 0 and b  = α  (Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)

(iv) Check that I-b is in the stabiliser of x. Also, show that there are infinitely many choices of a for which I - b is invertible.

Which of the following statements are true and which are false? Give reasons for your answer.

(a) If a finite group G acts on a finite set S, then G_s1 = G_s2 for all s₁, $₂ ∈ X.

(b) There are exactly 8 elements of order 3 in S₄.

(c)

(d)

(e) For any

Evaluate the following integrals

a) 

b)  

Show that there is only one linear fractional transformation that maps three given distinct points, and in the extended  plane onto three specified distinct points , and w,₃ in the extended w plane. 

Find the image of the semi-infinite strip  when . Sketch the strip and its image. 

22-1 Prove that the linear fractional transformation maps the circle  into itself. Also prove that  is conformal in 

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

Expand  in a Laurent series valid for

i)     ii) 

Find the radius of convergence of the following series.

i)   ii)        

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

Find the maximum modulus  on the closed circular region defined by

Evaluate de where e is the circle

Find all the singularities of the function