MATHS302 Final Assignment
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MATHS302 Final Assignment
1. Prove that in a finite 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 . Define the equaliser E(θ, π) to be the set of elements of G on which the two homomorphisms have the same effect. 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 defines a homomorphism from D4 into S4 . Differ- ent labellings will give different homomorphisms. Let θ and π be the homomorphisms obtained from the labellings in the diagram.
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θ |
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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 different 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.
2022-06-15