Please make sure that your name is on everything you hand in.

This is a ``closed book'' exam.

Answer as many questions as you can.

All questions have about the same number of marks.

1.

Find the highest common factor of 7 and 61 and express it as 7m+61n for some integers m and n.

2.

Prove that there is an infinite number of primes.

3.

Let G be the dihedral group of order 12 consisting of all symmetries of a regular hexagon. Find the conjugacy classes in G.

4.

Let U_n be the group of all elements [m] of Z_n such that m is coprime to n, with the group operation given by multiplication. (Recall that [m] is the element m+nZ of Z_n, the set of integers modulo n.) Find the orders of all the elements of U₁₅. Is U₁₅ cyclic?

5.

Let K be the subgroup of order 2 of G = U₁₅ consisting of the elements [1] and [4]. Write down all the cosets of K in G. Find an isomorphism from G/K onto Z₂×Z₂ (the product of two groups of order 2).