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- (a) List all the abelian groups of size 24 up to isomorphism (no proof required).
Show that none of the groups in your list is isomorphic to any other one.
(b) Give an example of a finite non-abelian group G such that |G| > 100 and x^6 = e for all x ∈ G.
(c) Write down all the elements of the dihedral group D 2 n in terms of a rotation ρ of order n and a reflection σ. Write down also an equation satisfied by ρ and σ which determines the multiplication table of D 2 n. (No proof required, just the equation.)
(d) Prove that if the alternating A 5 has a subgroup which is isomorphic to the dihedral group D 2 n, then n ≤ 5.
(e) Show that A 5 has a subgroup which is isomorphic to D 10.
(f) Let G be a group with the following properties:
|G| = 6 and G is non-abelian G has an element x of order 3, and an element y of order 2. Prove that G ∼= D 6.
- (a) Let G and H be groups. Define what is meant by
a homomorphism φ : G → H a normal subgroup N of G the factor group G/N
(b) Find all the normal subgroups N of the dihedral group D 8 such that |N | = 2. Justify your answer.
(c) Find, up to isomorphism, all groups H such that |H| = 4 and there is a surjective homomorphism φ : D 8 → H. (You may use any results you need from the course, provided you state them clearly.)
(d) Calculate the number of distinguishable necklaces that can be made from seven beads, four of which are red, two of which are blue, and one of which is yellow.
- (a) Define what is meant by the statement that two n × n matrices A and B are similar to each other.
(b) Let n ∈ N and λ ∈ C. Define the Jordan block Jn(λ).
State the Jordan Canonical Form theorem.
(c) Calculate the total number of non-similar Jordan canonical forms having characteristic polynomial x^4 (x − 1)(x + i)^3. Give your reasoning.
(d) Define 4 × 4 matrices A, B, C, D over C as follows:
A =
,^ B^ =
C =
,^ D^ =
Determine which pairs of these matrices are similar to each other. Give your reasoning.
