Some group tables and group computations
The two groups of order 4 (up to isomorphism): (i) Z/4Z:
0
0
1
2
3
+
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
Aside from the trivial subgroup, Z/4Z has one proper subgroup of order
2: 2 .
(ii) The Klein 4-group V (iso
Modern Algebra I Spring 2015
Review for the Second Midterm
Denition: Let G be a group. A subset H of G is a subgroup of G if H
is closed under the operation (i.e. for all h1 , h2 H, h1 h2 H), and the
binary structure (H, ) is a group. Note that the binary
Simple groups and the classication of nite groups
1
Finite groups of small order
How can we describe all nite groups? Before we address this question, lets
write down a list of all the nite groups of small orders 15, up to isomorphism. We have seen almost
The Sylow theorems
1
Denition of a p-Sylow subgroup
Lagranges theorem tells us that if G is a nite group and H G, then
#(H) divides #(G). As we have seen, the converse to Lagranges theorem
is false in general: if G is a nite group of order n and d divides
Cosets, Lagranges theorem and normal subgroups
1
Cosets
Our goal will be to generalize the construction of the group Z/nZ. The
idea there was to start with the group Z and the subgroup nZ = n , where
n N, and to construct a set Z/nZ which then turned out
Groups acting on sets
1
Denition and examples
Most of the groups we encounter are related to some other structure. For
example, groups arising in geometry or physics are often symmetry groups
of a geometric object (such as Dn ) or transformation groups of
Two subgroups and semi-direct products
1
First remarks
Throughout, we shall keep the following notation: G is a group, written
multiplicatively, and H and K are two subgroups of G. We dene the
subset HK of G as follows:
HK = cfw_hk : h H, k K.
It is not i
Modern Algebra I Spring 2015
Review Sheet for the Final
Denition: Let G1 and G2 be groups. A function f : G1 G2 is a
homomorphism if, for all g, h G1 , f (gh) = f (g)f (h).
Examples: 1) f an isomorphism, or more generally f an isomorphism from
G1 to a sub
MODERN ALGEBRA I SPRING 2016:
TENTH PROBLEM SET
1. Let f : Q Q be defined by f (r) = r/2 = ( 21 )r. Is f a homomorphism? Why or why not? Is f injective? Surjective?
(ii) Let g : Q Q be defined by g(r) = r/2 = ( 12 )r. Is g a homomorphism? Why or why not?
Numbers
1
Natural numbers
What are the basic properties of the natural numbers N? First, we need the
number 1. Second, given a number n N, we can always find a next number
which we will write as s(n) and think of as the successor of n. Note that 1 is
not
Cyclic groups and elementary number theory
1
Long division with remainder
The story of cyclic groups is very much connected with elementary number
theory (factorization, prime numbers, congruences). We prove some basic
facts from rst principles, using a f
Notes on the symmetric group
1
Computations in the symmetric group
Recall that, given a set X, the set SX of all bijections from X to itself (or,
more briey, permutations of X) is group under function composition. In
particular, for each n N, the symmetri
Equivalence relations
A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a
fraction a/b, a, b Z and b = 0, and hence specied by the pair (a, b)
Z (Z cfw_0). But diere
Numbers
1
Natural numbers
What are the basic properties of the natural numbers N? First, we need the
number 1. Second, given a number n N, we can always nd a next number
which we will write as s(n) and think of as the successor of n. Note that 1 is
not th
Review of linear algebra
1
Vectors and matrices
We will just touch very briey on certain aspects of linear algebra, most of
which should be familiar. Recall that we deal with vectors, i.e. elements of
Rn , which here we will denote with bold face letters
Sets and functions
1
Sets
The language of sets and functions pervades mathematics, and most of the
important operations in mathematics turn out to be functions or to be expressible in terms of functions. We will not dene what a set is, but take as
a basic
Binary operations and groups
1
Binary operations
The essence of algebra is to combine two things and get a third. We make
this into a denition:
Denition 1.1. Let X be a set. A binary operation on X is a function
F : X X X.
However, we dont write the value
Modern Algebra I Spring 2015
Review for the First Midterm
Sets and functions: Sets and are undened terms. For all x, X, either
x X or x X, but x X and x X never hold simultaneously. . Two
/
/
sets X and Y are equal for all x, x X x Y . There is a
unique s
Modern Algebra I: The Euclidean algorithm
As promised in the lecture, we describe a computationally ecient method
for nding the gcd of two positive integers a and b, which at the same time
shows how to write the gcd as a linear combination of a and b.
Beg
Subgroups and cyclic groups
1
Subgroups
In many of the examples of groups we have given, one of the groups is a
subset of another, with the same operations. This situation arises very often,
and we give it a special name:
Denition 1.1. A subgroup H of a g
Homomorphisms
1
Denition and examples
Recall that, if G and H are groups, an isomorphism f : G H is a bijection
f : G H such that, for all g1 , g2 G,
f (g1 g2 ) = f (g1 )f (g2 ).
There are many situations where we are given a function f : G H, which
is no
MODERN ALGEBRA I SPRING 2016:
THIRD PROBLEM SET
cos sin
cos sin
1. Let A =
and let B =
be 2 2
sin cos
sin cos
orthogonal matrices (depending ona real number
), with det A = 1
1 0
and det B = 1. Finally, let R =
.
0 1
(i) Show that every element of O2
MODERN ALGEBRA I SPRING 2016:
FOURTH PROBLEM SET
1. Let X be the set cfw_0, 1, 2, 3 and let be the operation on X defined
by a b = |a b|. Make a table for the operation . Show that
is commutative and that an identity and inverses exist for . Does
the Sud
Cosets, Lagranges theorem and normal subgroups
1
Cosets
Our goal will be to generalize the construction of the group Z/nZ. The
idea there was to start with the group Z and the subgroup nZ = hni, where
n N, and to construct a set Z/nZ which then turned out
MODERN ALGEBRA I SPRING 2016:
SEVENTH PROBLEM SET
1. Using the Euclidean algorithm or inspection, find the gcd of 40 and
51 and write the gcd as a linear combination of 40 and 51.
2. Do there exist integers x and y such that 57x + 93y = 2? Why or why
not?
MODERN ALGEBRA I SPRING 2016:
FIFTH PROBLEM SET
1. (i) In the group (Z/2Z) (Z/3Z), what is the order of (1, 1)? In the
group (Z/4Z) (Z/8Z), what is the order of (1, 1)? Of (2, 4)?
(ii) Let G and H be two groups and let g G and h H be two
elements of finit
The Sylow theorems
1
Definition of a p-Sylow subgroup
Lagranges theorem tells us that if G is a finite group and H G, then
#(H) divides #(G). As we have seen, the converse to Lagranges theorem
is false in general: if G is a finite group of order n and d d
Groups acting on sets
1
Definition and examples
Most of the groups we encounter are related to some other structure. For
example, groups arising in geometry or physics are often symmetry groups
of a geometric object (such as Dn ) or transformation groups
MODERN ALGEBRA I SPRING 2016:
EIGHTH PROBLEM SET
1. For the group Z/18Z, list all of the possible subgroups
P of Z/18Z together with all of their generators, and verify that d|18 (d) = 18.
2. (i) What is the order of the element 21 in the group Z/36Z? Wha
Modern Algebra I Spring 2016
Review for the First Midterm
Sets and functions: Sets and are undefined terms. For all x, X, either
x X or x
/ X, but x X and x
/ X never hold simultaneously. . Two
sets X and Y are equal for all x, x X x Y . There is a
uniq