Answer THREE questions

RESIT STUDENT (1 CREDIT): Answer TWO questions. Time: 1.5hrs

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.

2. Let G be a finite group acting on a finite set X. For x Î X let G(x) and G_x denote, respectively, the orbit and the stabiliser of x. Prove that

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|.

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. (a) Let S_n be the group of all permutations of the set {1,2,...,n}. For a Î S_n, let c(a) denote the number of cycles of a when it is written as a product of disjoint cycles. Let b Î S_n be a transposition. Give, without proof, the possible values of c(ba) in terms of c(a). Deduce from this that if a is written as a product of transpositions then the number of transpositions is always either odd or even.

(b) Let G be a subgroup of S_n, and let H be the subgroup of G consisting of all the even permutations in G. Show that either H = G or else |H| = |G|/2. (Hint: Define a suitable homomorphism from G to {1,-1}.)

Deduce that A_n, the group of all even permutations in S_n, has order equal to |S_n|/2.

(c) In the sequel you may assume that A₅ is simple.

Let G be a non-trivial normal subgroup of S₅, and let H be the subgroup of G consisting of all even permutations in G, that is, H = GÇA₅.

• Show that H is a normal subgroup of A₅, therefore H = {id} or H = A₅.
• Suppose G ¹ H, that is, G contains odd permutations. Show that, since G is a non-trivial normal subgroup of S₅, H = {id}, therefore G = {id,(ij)} for some transposition (ij). But then show that G is not a normal subgroup of S₅.
• Deduce that G = A₅.