Answer THREE questions

Now suppose |X| £ 90 and suppose G is a 7-group acting on X and having exactly one fixed point. Suppose also that H is an 11-group acting on X and that the action of H has no fixed points. Find |X|.

2. (a) Let G be a finite group, H £ G and X the set of left cosets of H in G. Show that there is an action of G on X such that the kernel of this action is contained in H.

Suppose G is a group of order 70 and suppose also that G contains a subgroup of order 14. Show that G cannot be simple.

(b) State carefully the three Sylow Theorems.

Prove that a group of order 992 cannot be simple.

3. 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 necklace is to be made from 9 beads strung on a circular wire; 6 of these beads are to be coloured white and 3 beads are to be coloured black. Ignoring the positioning of the fastening, how many essentially different necklaces can be made this way?

4. Obtain the class equation for a finite group, explaining clearly the terms conjugacy, centre and conjugacy class. Explain also why the order of a conjugacy class divides the order of the group.

Let G be a group of order 24 with centre consisting only of the identity element. Show that G has a conjugacy class of size 3, and deduce that G has a subgroup of order 8.

[You may use the Orbit-Stabiliser Theorem in this question, but Sylow's Theorems may not be used.]