1. State, with reasons, whether the following statements are True or False:

Any subgroup H c G of a group G is the kernel of a suitable group homomorphism

If m n 1 are natural numbers with m there is a group'G of order m and a set S with n elements such that G operates transitively on S.

For every n E n there are infinitely many irreducible polynomials of degree n over Q.

Let R1, R2 G GLn be two irreducible representations of the finite group G. Then, their characters xRI and xR2 are orthogonal only if R1 and R2 are inequivalent.

The fields and are isomorphic.

Solve the set of congruences

x 2 (mod

3 (mod

x 4 (mod 11)

simultaneously.

Show that L is a regular language.

Let a be the real cube root of 2 and let B be another (complex) root of X^3 0. Show that the fields and are isomorphic.

Let cr 2 3 8 (10 11) € A11. Check whether the conjugacy class of cr in Su splits into two conjugacy classes in A11 or not.

Calculate the Legendre Symbol(15/71)

Does X^P — X have a multiple root over Zp

Let p be an odd prime and
X^P 1. Then, show that all roots of f over Zp are multiple roots.

Check whether there is a group of order with class equation 1 2 4 5.

Prove that the subgroup SO2 of SU2 is conjugate to the subgroup T of diagonal vectors.

Show that cos n/8 is an algebraic number.

Let G be an abelian group of order n and suppose p is a prime such that n and 1+n. Let H be the Sylow p-group of G and m n/p^k". Show that, for any a € a^m€H

Show that any field extension of degree 2 is normal.

If G is a group of order then show that the number of inequivalent irreducible representations ofG is at most n.

Find the elementary divisors of the group
Z4 x Z6 x Z21 x Z35.

Determine the last row of the following character table of a group G of order 12 which has 4 conjugacy classes:

1 3 4 4
x1 x2 x3 x4
X1 1 1 1 1
X2 1 1 w^2 w
X3 1 1 w w^2
X4

Show that Q(sqrt(3)