Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Solutions to Quiz 1
1. Consider the integers a = 18 and b = 27.
(i) Find d = gcd(18, 27) and = lcm(18, 27).
d = 9,
= 54
(ii) What is the relationship between a, b, d, and
this relationship holds in th
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Solutions to Practice Quiz 4
1. Write down all the automorphisms of the group Z5 .
The automorphisms are k : Z5 Z5 , with k (x) = kx, for k = 1, 2, 3, 4.
2. Let R+ be the multiplicative group of posit
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Solutions to Practice Quiz 6
1. Let H be set of all 2 2 matrices of the form
ab
, with a, b, d R and
0d
ad = 0.
(a) Show that H is a subgroup of GL2 (R).
By denition, H is the set of nonsingular (inve
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 3, Solutions
Name:
1. (a) Find the conjugate of (1234)(56) by a = (25) in S7 .
Denition: A conjugate of by a is a ( ) = aa1 .
Remark: If order of an element a in a group is |a| = n, then a1 = an1 .
If a = (25)
MTH 3175 Group Theory (Prof.Todorov)
Quiz 2 (Practice)
Name:
Please explain all your work ! When using theorems, write their statements.
1. Let G be a group and let H and K be subgroups of G. Prove that H K is a subgroup of G.
2. Let G = U (20). Is G cycl
MTH 3175 Group Theory (Prof.Todorov)
Quiz 2 (Practice)
Name:
Please explain all your work ! When using theorems, write their statements.
1. Let G be a group and let H and K be subgroups of G. Prove that H K is a subgroup of G.
Proof: Let H and K be subgro
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 3 Practice
Name:
Please explain all your work ! When using theorems, write their statements.
1. (a) Find the conjugate of (1234)(56) by a = (25) in S7 .
(b) Find the conjugate of (1234)(56) by a = (27) in S7 .
S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice (Some Solutions) Name:
1. Consider external direct product: Z6 Z15 .
(a) What is the order of (2, 3) Z6 Z15 ?
Answer:
Order of (a, b) is |(a, b)| = lcm(|a|, |b|).
|(2, 3)| = lcm(|2|, |3|) = lcm(3,
S11MTH 3175 Group Theory (Prof.Todorov)
Quiz 4 Practice
Name:
1. Consider external direct product: Z6 Z15 .
(a) What is the order of (2, 3)?
(b) What is the order of (2, 12)?
(c) What is the order of (4, 12)?
(d) What is all possible orders of elements (a
S11MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 Practice
Name:
Some problems are really easy, some are harder, some are repetitions.
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even permutations. P
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 6 (PracticeSolutions)
Name:
Some of the problems are very easy, some are harder.
1. Let G and H be two groups and G H the external direct product of G and H .
(a) Prove that the map f : G H H G dened as f (g, h
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 6 (Practice)
Name:
Some of the problems are very easy, some are harder.
1. Let G and H be two groups and G H the external direct product of G and H .
(a) Prove that the map f : G H H G dened as f (g, h) = (h, g
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 (Practice Solutions)
Name:
Some problems are really easy, some are harder, some are repetitions.
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even per
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Quiz 3
1. (a) Draw the subgroup lattice of Z30 .
(b) Make a table with all the elements of Z30 , grouped according to their orders; how
many elements of each possible order are there?
2. Let a be an e
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Practice Quiz 6
1. Let H be set of all 2 2 matrices of the form
ab
, with a, b, d R and
0d
ad = 0.
(a) Show that H is a subgroup of GL2 (R).
(b) Is H a normal subgroup of GL2 (R)?
2. Let H = cfw_(1),
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Quiz 1
1. Consider the integers a = 18 and b = 27.
(i) Find d = gcd(18, 27) and
= lcm(18, 27).
(ii) What is the relationship between a, b, d, and
relationship holds in this situation.
predicted by the
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Practice Quiz 3
1. (a) Find the subgroup lattice of Z36 .
(b) Make a table with all the elements of Z36 , grouped according to their orders.
(c) What are all the possible orders, and how many elements
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
The dihedral groups
The general setup. The dihedral group Dn is the group of symmetries of a regular
polygon with n vertices. We think of this polygon as having vertices on the unit circle,
with verti
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Practice Quiz 1
1. Let d = gcd(20, 24).
(a) Find d.
(b) Find a pair of integers s and t such that 20s + 24t = d.
(c) Find the general solution for all the pairs of integers s and t such that 20s +24t
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Practice Quiz 2
1. Let G be a group and let H and K be subgroups of G.
(a) Is H K a subgroup of G?
(b) Is H K a subgroup of G?
In each case, give a reason why, or why not.
2. Let H be a subgroup of a
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Practice Quiz 4
1. Write down all the automorphisms of the group Z5 .
2. Let R+ be the multiplicative group of positive real numbers. Show that the map
x 3 x is an automorphism of R+ .
3. Show that th
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Some solutions to the problems on Practice Quiz 3
1. (a) Find the subgroup lattice of Z36 .
Subgroups: 1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 0
You should draw now the inclusions between these various subg
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Quiz 6
1. Let H be set of all 2 2 matrices of the form
a0
, with a, c, d Z and ad = 1.
cd
(a) Show that H is a subgroup of GL2 (Z).
(b) Is H a normal subgroup of GL2 (Z)?
2. Let G = U (16), and H = cf
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Solutions for Quiz 3
1. (a) Draw the subgroup lattice of Z30 .
1
AA
AA
AA
AA
2
3
CC
CC cfw_
C
cfw_
cfw_ CCC
cfw_
CC
CC cfw_
C
cfw_
cfw_ CCC
cfw_
5
6
10
CC
CC
CC
CC
cfw_
cfw_
cfw_
cfw_
0
15
(b) Ma
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Solutions to Quiz 5
1. List all the elements of Z2 Z8 , and compute their orders.
Element (a, b) (0,0) (1,0) (0,1) (1,1) (0,2) (1,2) (0,3) (1,3)
order |(a, b)|
1
2
8
8
4
4
8
8
Element (a, b) (0,4) (1,
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Quiz 5
1. List all the elements of Z2 Z8 , and compute their orders.
2. Show that the group U (9) is isomorphic to the direct product Z2 Z3 , by describing
explicitly an isomorphism : U (9) Z2 Z3 .
3.
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Quiz 6
1. Let H be set of all 2 2 matrices of the form
a0
, with a, c, d Z and ad = 1.
cd
(a) Show that H is a subgroup of GL2 (Z).
H is a subset of GL2 (Z) and
The identity
10
H.
01
Closed under m