Table of Contents
Things Past
Roots of Unity
Every complex number 
has a factorization
1
where 
and 
If 
and 
, then
2
3
If 
, an is a complex number 
with
Every nth root of unity is equal to
4
for 
5
If 
is an nth root of unity and if 
is the smallest, then is a
n
If 
, then the $d$th is
6
where ranges over all the primitive dth roots of unity
For every integer 
7
Define φas the degree of the nth cyclotomic
polynomial
8
If 
is an integer, then 
is the number of integers 
with
and 
Suffice to prove 
is a primitive nth root of unity if and only
if 
and 
are relatively prime
For every integer 
, we have
9
Group
Permutations
A of a set 
is a bijection from 
to itself.
The family of all the permutations of a set , denoted by 
is called
the on . When 
, 
is
usually denoted by 
and is called the
Let 
be distinct integers in 
. If
fixes the other integers and if
10
then α is called an textbf{r-cycle}. α is a cycle of
and denoted by
11
2-cycles are also called the transpositions.
Two permutations 
are if every 
moved by one is fixed by the other.
Disjoint permutations 
commute
Every permutation 
is either a cycle or a product of disjoint cycles.
Induction on the number of points moved by 
A of a permutation 
is a
factorization of into disjoint cycles that contains exactly one
1-cycle 
for every 
fixed by
Let and let 
be a complete
factorization into disjoint cycles. This factorization is unique except for
the order in which the cycles occur
for all , if 
, then 
for any 
If 
, then 
has the same cycle
structure as 
. In more detail, if the complete factorization of
is
12
then 
is permutation that is obtained from
by applying to the symbols in the cycles of
Example. Suppose
13
then we can easily find the
14
Permutations and 
in 
has the same cycle structure if
and only if there exists with 
If 
then every is a product of tranpositions
A permutation is if it can be factored into a
product of an even number of transpositions. Otherwise
If and 
is a complete
factorization, then is defined by
15
For all
16
- Let ; if
then is even. otherwise
odd - A permutation is odd if and only if it's a product of an odd
number of transpositions
Let . If and 
have the same parity, then
is even while if and 
have distinct parity,
is odd
Groups
A on a set 
is a function
17
A is a set 
equipped with a binary operation * s.t.
- the holds
- every
has an , there is a 
with
A group is called if it satisfies the
Let be a group
- The holds: if either
or 
, then
is unique- Each
has a unique inverse 
An expression 
if all the ultimate
products it yields are equal
If is a group and 
then the expression
needs no parentheses
Let be a group and let 
. If 
for some 
then the
smallest such exponent 
is called the or 
; if no such
power exists, then one says that 
has
If is a finite group, then every has finite order
A is a distance preserving bijection 
. If
π is a polygon in the plane, then its 
consists of all the motions 
for which 
. The
elements of 
are called the of π
Let 
be a square. Then the group 
is called the
with 8 elements, denoted by 
If 
is a regular polygon with vertices 
and center
, then the symmetry group 
is called the {dihedral
group} with 
elements, and it's denoted by 
Lagrange's theorem
A subset 
of a group is a if
- if
, then 
- if
, then 
If 
is a subgroup of , we write 
. If is a proper subgroup,
then we write $H<$G
A subset of a group is a subgroup if and only if is nonempty and
whenever 
, 
A nonempty subset of a finite group is a subgroup if and only if
is closed; that is, if 
, then 
If is a group and 
18
is called the of by . A
group is called if there exists s.t. 
,
in which case is called the
The 
, denoted by 
is the family of all congruence
classes mod 
Let 
be a fixed integer
- If
, then 
for some 
with 
- If
, then 
has exactly 
elements
- If
is a cyclic group of order , then 
is a generator
of if and only if 
-
If
is a cyclic group of order and 
, then
19
where
is the Euler φ-function
- there is
s.t. 
hence 
and 
Let be a finite group and let . Then the order of is
.
If is a finite group, then the number of elements in , denoted by
is called the of
The intersection 
of any family of subgroups of a group
is again a subgroup of
If is a subset of a group , then there is a subgroup 
of
containing tHhat is in the sense that 
for
every subgroup
of that contains
If is a subset of a group , then 
is called the {subgroup
generated by}
A on is an element 
of the form 
where 
and 
for all
If is a nonempty subset of a group , then is the set of all
words on
If 
and , then the 
is the subset 
of ,
where
20
, 
if and only if 
- if
, then 
for all
define a relation 
if 
If is a subgroup of a finite group , then 
is a divisor of 
Let 
be the family of all the distinct cosets of
in . Then
21
hence
22
But 
for all . Hence 
The of a subgroup in denoted by 
, is the number of
left cosets of in
Note that 
If is a finite group and , then the order of is a divisor of
If is a finite group, then 
for all
If 
is a prime, then every group of order 
is cyclic
The set 
, defined by
23
is a multiplicative group of order 
. If is a prime, then
, the nonzero elements of 
.
If is a prime and 
, then
24
suffices to show 
in 
. If 
, then 
.
Else, since 
, 
If 
, then
25
An integer is a prime if and only if
26
Homomorphisms
If 
and 
are groups, then a function 
is a
if
27
for all 
. If 
is also a bijection, then 
is called an
. and are called , denoted by 
Let 
be a homomorphism
for all 
If is a homomorphism, define
28
and
29
Let be a homomorphism
is a subgroup of and 
is a subgroup of - if
and if , then 
- is an injection if and only if
A subgroup 
of a group is called a if 
and 
imply 
, denoted by 
If is a group and , then a of is any element
in of the form
30
where
If is a group and , define 
by
31
for all
- If is a group and , then conjugation
is an
isomorphism - Conjugate elements have the same order
- bijection:
- If is a subgroup of index 2 in a group , then
for every
- If is a subgroup of index 2 in a group , then is a normal
subgroup of
The group of is the group 
of order 8 consisting of the
following matrices in 
32
where 
is the identity matrix
33
The alternating group 
is a group of order 12 having no subgroup of
order 6
Quotient group
is the set of all nonempty subsets of a group . If
, define
34
is normal if and only if
35
A natural question is that whether 
is a subgroup when and 
are
subgroups. The answer is no. Let 
- If and are subgroups of a group , and if one of them is normal,
then
and 
- If
, then 
Let 
denote the family of all the left cosets of a subgroup of .
If 
, then
36
for all 
and 
is a group under this operation
is called the mod
Every 
is the kernel of some homomorphism
Define the 
, 
If is a homomorphism, then
37
If 
and 
, then 
is an isomorphism
38
If and are subgroups of a finite group , then
38
Define a function 
. Show that
.
Claim that if 
, then
39
If 
, then 
and
40
If 
with 
, then 
and
41
If 
is the natural map, then
42
is a bijection between 
, the family of all those subgroups 
of
that contain , and 
, the family of all the subgroups of
. If we denote 
by 
, then
if and only if 
, in which case 
if and only if 
, in which case 
43
Use 
and 
to prove injectivity and surjectivity
respectively.
For 
, show there is a bijection between the family of all
cosets of the form 
and the family of all the cosets of the form
.
injective:
43
surjective:
If is finite, then
44
If 
, by third isomorphism theorem, 
If 
,
45
so that 
If is a finite abelian group and 
is a divisor of , then
contains a subgroup of order 
Abelian group's subgroup is normal and hence we can build quotient groups.
p90 for proof. Use the correspondence theorem
If and are grops, then their , denoted by 
, is the set of all ordered pairs 
with the operation
46
Let and 
be groups and 
. Then 
and
47
If is a group containing normal subgroups and and 
and 
, then 
Note 
. Consider 
. Show it's homo and bijective.
If 
are relatively prime, then
48
Let be a group, and 
be commuting elements of orders 
. If
, then 
has order 
If 
, then 
- If is a prime, then
-
If
, then
49
A cyclic group of order has a unique subgroup of order , for each
divisor of , and this subgroup is cyclic.
Define an equivalence relation on a group by 
if 
. Denote the equivalence class containing 
by 
, where 
. Equivalence classes form a partition and we get
50
where 
ranges over all cyclic subgroups of . Note 
A group of order is cyclic if and only if for each divisor of
, there is at most one cyclic subgroup of order
If is an abelian group of order having at most one cyclic subgroup o
f
order for each prime divisor of , then is cyclic
Exercise:
- 2.71 Suppose
. Since 
,
. Hence 
- 2.67 1.
and 
.
Hence
Group Actions
Every group is isomorphic to a subgroup of the symmetric group 
. In
particular, if 
, then is isomorphic to a subgroup of 
For each , define 
for every . 
is a
bijection for its inverse is 
51
Let be a group and having finite index . Then there exists a
homomorphism 
with 
When 
, this is the Cayley theorem.
Every group of order 4 is isomorphic to either 
or the four-group
. And 
By lagrange's theorem, every element in other than 1 has order 2 or 4. If
4, then is cyclic.
Suppose 
, then 
. Hence 
.
If is a group of order 6, then is isomorphic to either 
or
. Moreover 
If is not cyclic. Since is even, it has some elements having
order 2, say 
.
If is abelian. Suppose it has another different element with order 2.
Then 
is a subgroup which contradict. Hence it must contain
an element 
of order 3. Then 
has order 6 and is cyclic.
If is not abelian. If doesn't have elements of order 3, then it's
abelian. Hence has an element 
of order 3.
Now 
, so 
and 
is normal. Since 
, 
. If 
, 
.
If 
, 
. If 
.
Let 
, 
given by
52
By representation on cosets, 
. Hence
or 
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