School of Mathematics and Statistics
MT4003 Groups
Problem Sheet VIII: The Centre, Commutators and Conjugation
数学作业代写价格 Let G be a non-abelian simple group. Show that every conjugacy class of G, apart from {1}, contains at least two elements.
1.Let G be a group and H be a subgroup of G such that H 6 Z(G). Show that H is a normal subgroup of G. 数学作业代写价格
2.(a) Show that centre of Sn is trivial for n > 3.
(b) Show that the centre of the dihedral group D2n is trivial if n is odd and is cyclic of order 2 if n is even.
(c) Show that the centre of An is trivial for n > 4.
(d) What is Z(A3)?
3.Let p be a prime number. Show that
Z(GL2(p)) = { αI | α∈ Fp, α ≠ 0 },
where I denotes the identity matrix.
4.Prove that if H is a subgroup of a group G such that G‘ ≤ H, then H is a normal subgroup of G.
5.Let G and H be groups. Prove that (G × H)‘ = G’ × H’ . 数学作业代写价格
[Hint: Prove that every generator of each of these groups belongs to the other.]
6.Find A’n and S’n for each n > 1.
[Hint: You should consider di↵erent possibilities for n. For example, n ≥ 5 corresponds to dierent behaviour to n = 4, etc.]
7.List the conjugacy classes of the symmetric group S5, giving one element from each conjugacy class and the size of each conjugacy class.
What is the order of the centraliser CS5 ((1 2 3 4 5))?
How many conjugates does (1 2 3 4 5) have in A5?
Is (1 2 3 4 5) conjugate to (1 2 3 5 4) in A5?
8.Find the sizes of the conjugacy classes in S4.
Hence determine all normal subgroups of S4.
9.Let n be odd and let α and β denote the usual generators of the dihedral group D2n of order 2n. 数学作业代写价格
Find the centralisers of the elements β and αi (where 0 ≤ i ≤ n/2).
Hence determine the conjugacy classes of D2n when n is odd.
What are the conjugacy classes of D2n when n is even?
10.Let G be a non-abelian simple group. Show that every conjugacy class of G, apart from {1}, contains at least two elements.
11.A fifinite p-group is a fifinite group whose order is a power of the prime number p.
Use the Class Equation to show that if G is a non-trivial p-group, then the centre Z(G) of G is non-trivial. 数学作业代写价格
[Hint: If an element xi lies in a conjugacy class of size > 1, then the centraliser CG(xi) is a proper subgroup of G. What can you say about its index?]
12.Let p be a prime number. Show that a fifinite p-group is simple if and only if it is cyclic of order p.
