MA 573-001
HOMEWORK SHEET 1
Due Tuesday September 8, 2010
1. Let G be a group and suppose that x2 = 1 for all x G. Prove
that G is abelian.
2. Prove that a group G has exactly three subgroups if and only
if the group is cyclic of order p2 for some prime p
MA 573-001
HOMEWORK SHEET 2
Due Tuesday September 14, 2010
1.
(i) Suppose that Sn and that = 1 2 . . . k as a product of
disjoint cycles. Suppose that i has nite order mi . Prove that has
order equal to lcm (m1 , . . . , mk ).
(ii) Find the distinct subgr
MA 573-001
HOMEWORK SHEET 3
Due Tuesday September 21, 2010
1. Suppose that G is a group and H G. Let X = cfw_gH | g G
be the set of left H -cosets. Prove that there is a homomorphism :
G Sym X given by (g ) = g where g (xH ) = gxH. Show that the
kernel of
MA 573-001
HOMEWORK SHEET 4
Due Tuesday September 28, 2010
1. Let G be the dihedral group of order 2n dened as G = a, b|an =
b2 = 1, b1 ab = a1 . Find Z (G) and G , the center and derived
subgroup respectively. Find also the order of each element of the f
MA 573-001
HOMEWORK SHEET 5
Due Tuesday October 5, 2010
1. Let G be a nite group and suppose that p is the least prime
dividing |G|. Suppose that H G and that |G : H | = p. Prove that
H G.
2. Prove that if x, y are two elements of a group G which lie in
t
MA 573-001 Solutions
HOMEWORK SHEET 1, Some
1. Let G be a group and suppose that x2 = 1 for all x G. Prove that G is abelian. Let x, y G. Then 1 = (xy )2 = (xy )(xy ). So y 1 x1 = y 1 x1 (xy )(xy ) = xy . However y 1 = y and x1 = x so we get yx = xy and t
MA 573-001 Sketch Solutions
HOMEWORK SHEET 2-Some
1. (i) Suppose that Sn and that = 1 2 . . . k as a product of disjoint cycles. Suppose that i has finite order mi . Prove that has order equal to lcm (m1 , . . . , mk ). (ii) Find the distinct subgroups of
MA 573-001 solutions
HOMEWORK SHEET 3-Some
1. Suppose that G is a group and H G. Let X = cfw_gH | g G be the set of left H -cosets. Prove that there is a homomorphism : G Sym X given by (g ) = g where g (xH ) = gxH. Show that the kernel of is precisely cf