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 relation
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,
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
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Solutions to Quiz 2
1. Let G be the group dened by the following Cayley table.
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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 elemen
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 th
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 th
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
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
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
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, .
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
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) P
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
Name:
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Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Final Exam
1. Let G be the group dened by the following Cayley table.
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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 elem
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)
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
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)
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 po
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 fo
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
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 t
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 sh
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 no
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
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3
CC
CC cfw_
C
cfw_
cfw_ CCC
cfw_
CC
CC cfw_
C
cfw_
cfw_ CCC
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)
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
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
Prof. Alexandru Suciu
MATH 3175
Group Theory
Fall 2010
Quiz 2
1. Let G be the group dened by the following Cayley table.
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