MATHS302 Final Assignment

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MATHS302 Final Assignment

1. Prove that in a nite group the number of elements of order 5 is a multiple of 4.

2. Let θ and π be two homomorphisms from G to H . De ne the equaliser E(θ, π) to be the set of elements of G on which the two homomorphisms have the same e ect. That is E(θ, π) = {g e G : θ(g) = π(g)}.

(a) Prove that E(θ, π) is a subgroup of G.

(b) Consider the dihedral group D4 which gives the symmetries of a square. Labelling the corners de nes a homomorphism from D4 into S4 . Di er- ent labellings will give di erent homomorphisms. Let θ and π be the homomorphisms obtained from the labellings in the diagram.

θ

π

List the elements of E(θ, π) in this case. Is it normal

3. A stick model of an octahedron is to be made using coloured plastic straws.

If there are six colours of straw available, how many rotationally distinct mod- els can be made

4. Show that S4 is soluble but not nilpotent.

5. If α and β are two di erent permutations in Sn explain why αβ and βα must have the same cycle structure.

6. Let H 司 G/N . Show that there is a normal subgroup K 司 G with H = K/N .

7. Prove that there are no simple groups of these orders.

(a) lGl = 400

(b) lGl = 1024

(c) lGl = 1452

8. Up to isomorphism, how many abelian groups of order 900 are there.

9. If H < G show that there is a Sylow p-subgroup P < G so that P n H is a Sylow p-subgroup of H . 10. Show that all groups of order 77 are cyclic.