Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 1 Due January 17th, 2014, in-class
1. Recall that Dn , the Dihedral group on n elements, is the set of all ips and rotations on a polygon of
n-elements. Dene D to be the set of all ips a

PMATH 336
Review Problems on Burnsides Theorem
Solutions
1. A Waterloo start-up company sells hexagonal oor tiles over the internet. Each tile has a white
centre, while the border is designed so that each edge can have one of three colours (blue, red,
gre

PMATH 336
Review Problems on Burnsides Theorem
Winter 2013
1. A Waterloo start-up company sells hexagonal oor tiles over the internet. Each tile has a white
centre, while the border is designed so that each edge can have one of three colours (blue, red,
g

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Signature
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PMath 336
Term Test
Friday March 7, 2008
1. [7 points] Let G be a group of permutations on a set S and let s S. Dene StabG (s),
OrbG (s) and state the stabilizer-orbit theorem.
Solution:
StabG (s) = cfw_g G | g(

PMATH 336 Second Midterm Test, 16 March 2007
55 minutes
[4]
1. (a) Carefully complete the following denition:
Let G = (G, ) and G = (G, ) be groups. An isomorphism from G to G is . . .
[12]
(b) For each of the following pairs of groups: (i) state whether

PMATH 336 First Midterm Test, 9 February 2007
55 minutes
[5]
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 1 in terms of a, b, c.
0 1
.
1
1
[4]
(b) In GL(2, R), determ

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 4 Due March 7th, 2014, in-class
1. Classify all nite rotation groups in R3 of size 102.
2. Describe the Frieze group of the following innite sequences.
(a)
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 5 Due March 21st, 2014, in-class
1. (a) Show that (2, 2)
ZZ
(b) Determine the order of Z Z/ (2, 2) .
(c) Determine the group Z Z/ (2, 2) .
2. Show that Z(G)
G.
3. Let C 0 be the space of

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 1 Due January 17th, 2014, in-class
1. Recall that Dn , the Dihedral group on n elements, is the set of all ips and rotations on a polygon of
n-elements. Dene D to be the set of all ips a

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 3 Due February 14th, 2014, in-class
1. How many elements of order 6 are there in S5 ? In S6 ?
2. Suppose that H and K are subgroups of G. Further, suppose that there exists a, b G such t

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 2 Due January 31st, 2014, in-class
1. Draw the subgroup lattice of Zpq2 for p and q distinct primes.
2. Consider the alternating permutation group A8 .
(a) Find a cyclic subgroup of A8 o

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 2 Due January 31st, 2014, in-class
1. Draw the subgroup lattice of Zpq2 for p and q distinct primes.
2. Consider the alternating permutation group A8 .
(a) Find a cyclic subgroup of A8 o

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 3 Due February 14th, 2014, in-class
1. How many elements of order 6 are there in S5 ? In S6 ?
2. Suppose that H and K are subgroups of G. Further, suppose that there exists a, b G such t

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 1 Due January 17th, 2014, in-class
1. Recall that Dn , the Dihedral group on n elements, is the set of all ips and rotations on a polygon of
n-elements. Dene D to be the set of all ips a

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 2 Due January 31st, 2014, in-class
1. Draw the subgroup lattice of Zpq2 for p and q distinct primes.
1
p
q
q2
pq
pq 2
Figure 1: Subgroup lattice for Zpq2
2. Consider the alternating perm

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 3 Due February 14th, 2014, in-class
1. How many elements of order 6 are there in S5 ? In S6 ?
We see elements of order 6 in S5 must be of the form (a, b, c)(d, e). We see that there are

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 4 Due March 7th, 2014, in-class
1. Classify all nite rotation groups in R3 of size 102.
2. Describe the Frieze group of the following innite sequences.
(a)
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 4 Due March 7th, 2014, in-class
1. Classify all nite rotation groups in R3 of size 102.
All nite rotations are Dn , Zn , A4 , A5 or S4 . If it is size 102, then it is clearly not any of

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 5 Due March 21st, 2014, in-class
1. (a) Show that (2, 2)
ZZ
(b) Determine the order of Z Z/ (2, 2) .
(c) Determine the group Z Z/ (2, 2) .
2. Show that Z(G)
G.
3. Let C 0 be the space of

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 6 Due April 4th, 2014, in-classs
1. Let G be a nite abelian group of order 105. How many elements does G have of orders 1, 3, 5, 7, 15, 21, 35
and 105?
2. Show that if a cl(b) then b cl(

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 5 Due March 21st, 2014, in-class
1. (a) Show that (2, 2)
ZZ
(b) Determine the order of Z Z/ (2, 2) .
(c) Determine the group Z Z/ (2, 2) .
(a) All subgroups of abelian groups are normal.

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 6 Due April 4th, 2014, in-classs
1. Let G be a nite abelian group of order 105. How many elements does G have of orders 1, 3, 5, 7, 15, 21, 35
and 105?
We see that G is a nite abelian gr

Pure Math 336, (Winter 2014) Introduction to Group Theory
Assignment 6 Due April 4th, 2014, in-classs
1. Let G be a nite abelian group of order 105. How many elements does G have of orders 1, 3, 5, 7, 15, 21, 35
and 105?
2. Show that if a cl(b) then b cl(