PMATH 346
Assignment 1
Winter 2013
Due Wednesday, January 23, 2013 by 9:20 a.m.
1. Let G = cfw_a, b, c, d.
(a) Show that there are exactly two ways that the following partial table can be completed so that
(G, ) is a group:
ab
aa
b
a
c
d
c
d
(b) Prove tha
PMATH 346
Assignment 6
Winter 2013
Due Wednesday, March 13, 2013 by 9:20 a.m.
1. Let G be a group. Recall that Z (G) = cfw_x G : xg = gx for all g G.
(a) Prove that Z (G)
G.
(b) Prove that if G/Z (G) is cyclic, then G is abelian.
(c) Prove that every grou
PMATH 346
Assignment 7 Revised
Winter 2013
Due Wednesday, March 20, 2013 by 9:20 a.m.
1. Prove that every group of order 1225 is abelian.
2. (a) Suppose G is a group of order pm where p is prime and p | m. Let np be the number of Sylow
p-subgroups of G. E
PMATH 346
Assignment 5
Winter 2013
Due Wednesday, February 27, 2013 by 9:20 a.m.
1. Dene G and H as follows:
G=
ab
0d
: a, b, d R, ad = 0
and H =
1b
01
: bR .
(a) Prove that H G GL(2, R).
(b) Prove that H
G but H
GL(2, R).
(c) Determine whether or not the
PMATH 346
Assignment 4
Winter 2013
Due Wednesday, February 13, 2013 by 9:20 a.m.
1. Given a group G, let A = G and for g, a G dene g a = gag 1 .
(a) Prove that this (G, G, ) is an action. (It is called the action of G on itself by conjugation.)
10
. With
PMATH 346
Assignment 2
Winter 2013
Due Wednesday, January 30, 2013 by 9:20 a.m.
1. Let G be an abelian group. Fix n Z.
(a) Prove that the set cfw_a G : an = 1 is a subgroup of G.
(b) Prove that the set cfw_an : a G is a subgroup of G.
2. If (A, ) and (B,
PMATH 346
Assignment 3
Winter 2013
Due Wednesday, February 6, 2013 by 9:20 a.m.
1. (a) Suppose G is a nite group with |G| = 20. Prove that if a G satises a4 = 1 and a10 = 1,
then a = G.
(b) Generalize.
2. Given a group G, its centre is the set Z (G) = cfw
PMATH 346
Assignment 8
Winter 2013
Due Wednesday, March 27, 2013 by 9:20 a.m.
1. (a) Suppose G is a group and H
of G.
G. Prove that H is a union of (some of the) conjugacy classes
(b) Prove that A5 is simple. [Hint: the class equation; see Assignment 7.]
PMATH 346
Assignment 9 Revised
Winter 2013
Due Friday, April 5, 2013 by 9:20 a.m.
1. How many abelian groups of order 4,000 are there (up to isomorphism)?
2. Suppose G is a nite abelian group. Show that if d is a divisor of |G|, then G has a subgroup of
o
PMATH 346
Assignment 6 - Solutions
Winter 2013
1. Let G be a group. Recall that Z (G) = cfw_x G : xg = gx for all g G.
(a) Prove that Z (G)
G.
Solution. We know Z (G) G (Assignment 3). Let g G. Then for every x Z (G) we have
gxg 1 = xgg 1 (denition of x Z
PMATH 346
Midterm Test Solutions
Winter 2013
1. Instruction: For each group G listed below, state the number of elements in G of each possible
order. (I.e., how many elements of order 1, how many elements of order 2, etc.) No justication is
required.
[4]
PMATH 346
Assignment 5 Solutions
Winter 2013
1. Dene G and H as follows:
ab
0d
G=
: a, b, d R, ad = 0
1b
01
and H =
: bR .
(a) Prove that H G GL(2, R).
Solution. If A =
ab
0d
AB 1 =
G and B =
1
xw
ab
0d
xy
0w
w y
0
x
w y
0
x
1
G, then B 1 = xw
=
1
xw
aw
PMATH 346
Assignment 4
Winter 2013
Due Wednesday, February 13, 2013 by 9:20 a.m.
1. Given a group G, let A = G and for g, a G dene g a = gag 1 .
(a) Prove that this (G, G, ) is an action. (It is called the action of G on itself by conjugation.)
Solution.
PMATH 346
Assignment 1
Winter 2013
Solutions
1. Let G = cfw_a, b, c, d.
(a) Show that there are exactly two ways that the following partial table can be completed so that
(G, ) is a group:
ab
aa
b
a
c
d
c
d
Solution: aa = a implies a is the identity, whic
PMATH 346
Assignment 2 Solutions
Winter 2013
1. Let G be an abelian group. Fix n Z.
(a) Prove that the set cfw_a G : an = 1 is a subgroup of G.
Solution. Use either the denition of subgroup (nonempty, closed under product, closed under
inverses) or the Su
PMATH 346
Assignment 3 Solutions
Winter 2013
1. (a) Suppose G is a nite group with |G| = 20. Prove that if a G satises a4 = 1 and a10 = 1,
then a = G.
Solution. |a| is a divisor of 20. Note that if |a| = 20 then |a| must be a divisor of either 4
or 10 (or