PMATH 336
Assignment 1
Winter 2013
Due Friday, January 25, 2013 by 3:20 p.m.
def
1. Consider the pair (Z, ) where a b = a + b 3.
(a) Carefully prove that (Z, ) is a group. What is the identity element of the group?
(b) Find 1 , the cyclic subgroup generat
PMATH 336
Assignment 6
Winter 2013
Due Monday, April 1, 2013 by 3:20 p.m.
1. Let G = Z5 D6 S5 .
(a) What is |G|?
(b) What is the identity element of G?
(c) If x = (2, 2 , (1 3 2 5), what is |x|?
2. Let G = Z4 A4 . Also let x = (3, (1 2)(3 4) and y = (1, (
PMATH 336
Assignment 7
Winter 2013
Due Friday, April 5, 2013 by 3:20 p.m.
1. Suppose is a group action of G on X . For a, b X , dene a b if and only if a = g b for some
g G. Prove that is an equivalence relation on X .
df
2. Suppose is a group action of G
PMATH 336
Assignment 5
Winter 2013
Due Friday, March 15, 2013 by 3:20 p.m.
1. Let G = GL(2, R) and let
H=
r0
0r
: r R
K=
r0
0s
: r, s R .
It is a fact, which you may assume, that H GL(2, R) and K GL(2, R).
(a) Prove that H
GL(2, R).
(b) Determine whether
PMATH 336
Assignment 4
Winter 2013
Due Friday, February 15, 2013 by 3:20 p.m.
1. (a) Show that the only subgroup of S3 containing (1 2 3) and (1 2) is S3 itself.
(b) Show that the only subgroup of S7 containing (1 2 3 4 5 6 7) and (1 2) is S7 itself. [Hin
PMATH 336
Assignment 2
Winter 2013
Due Friday, February 1, 2013 by 3:20 p.m.
1. Suppose G is a group and a, b G.
(a) If a2 = b2 and a3 = b3 , prove that a = b.
(b) Can you think of a way to generalize the claim in part (a)? Can you prove your generalizati
PMATH 336
Assignment 3
Winter 2013
Due Monday, February 11, 2013 by 3:20 p.m.
1. Let G be a nite group and p a prime number dividing |G|.
(a) Suppose H G with |H | = p. Prove that every element of H except e has order p.
(b) Suppose H, K G with |H | = |K
PMATH 336
Assignment 7 Solutions
Winter 2013
1. Suppose is a group action of G on X . For a, b X , dene a b if and only if a = g b for some
g G. Prove that is an equivalence relation on X .
Solution. Must show that is reexive, symmetric and transitive.
Re
PMATH 336
Midterm Test Solutions
Winter 2013
[3] 1. (a) Suppose G is a group and a, b, c, x, y are elements of G. If
ax = bc
ya = cx1 ,
nd an expression (as simple as possible) for y in terms of a, b, c.
Solution. y = (cx1 )a1 = c(ax)1 = c(bc)1 = c(c1 b1
PMATH 336
Assignment 6 Solutions
Winter 2013
1. Let G = Z5 D6 S5 .
(a) What is |G|?
Solution. |G| = |Z5 | |D6 | |S5 | = 5 12 120 = 7200.
(b) What is the identity element of G?
Solution. It is (eZ5 , eD6 , eS5 ), i.e., (0, e, 1) (where e is the identity el
PMATH 336
Assignment 5 Solutions
Winter 2013
1. Let G = GL(2, R) and let
H=
r0
0r
: r R
K=
r0
0s
: r, s R .
It is a fact, which you may assume, that H GL(2, R) and K GL(2, R).
(a) Prove that H
GL(2, R).
Solution. I will show that AHA1 H for all A GL(2, R)