Answer THREE questions

Now suppose that |G| = pq where p and q are distinct primes, and suppose also that |X| < p+q and that none of p,q divides |X|. Prove that the action of G on X has at least one fixed point.

2. Let G be a finite group acting on a finite set X. For each g Î G, let F(g) denote the set {x Î X: [^(g)](x) = x}, where [^(g)] denotes the permutation of X corresponding to g under the action.

Prove that the number of orbits in X under this action is given by

A 10×10 square grid is drawn on one side of a cardboard square. Four identical circles are drawn, each circle inside one of the cells of the grid. How many different grids can be drawn this way?

3. (a) You are given this information about the group G: G is not cyclic and it contains a subgroup H of order m (m an odd prime) and index 2. Describe fully the structure of G, explaining clearly your reasoning.

(b) Classify all groups of order 45.

[You may use without proving any of the Sylow Theorems and any results on p-groups and direct products. Any of these results which you do use should, however, be clearly stated.]

4. Let G be a group and H a subgroup of G. Let the index of H be m and let X be the set of all left cosets of H in G.

Show that there is an action of G on X whose kernel K is the largest normal subgroup of G contained in H.

Show also that |G|/|K| divides m!.

Suppose that |G| = 385 and |H| = 77. Prove that H is a normal subgroup of G.

[If required, the First Isomorphism Theorem may be quoted without proof, but its use must be clearly explained.]