is invertivble if there is a 
s.t.
Properties of Inverses
- If an element
has both a left inverse 
and a right inverse 
, i.e., if 
and 
, then 
, is inveritble, is its inverse - If is invertible, its inverse is unique
- Inverses multiply in the opposite order. If and
are invertible, so is the product 
, and 
. - An element may have a left inverse or a right inverse, though it is not invertible.
Group
A group 
is a set with a law of composition which is
(1) associative
(2) has an identity element
(3) every element is invertible
is a set with a law of composition which is(1) associative
(2) has an identity element
(3) every element is invertible
Cancellation Law
Let 
be elements of a group whose law of composition is written multiplicatively. If 
or if 
then 
. If 
or if 
, then b = 1.
be elements of a group whose law of composition is written multiplicatively. If or if then . If or if , then b = 1.Symmetric Group
The group of permutations of the set of indices 
is called the symmetric group and is denoted by 
.
is called the symmetric group and is denoted by .
) denoted 
(closed under LOC)
(identity)
(inverses)
Generators
generate if every element in can be expressed as a combination of 
's and 
's using LoC.Subgroups of 
Let 
be a subgroup of the additive identity 
. Either is the trivial subgroup 
, or else it has the form:
where is the smallest positive integer in .
be a subgroup of the additive identity . Either is the trivial subgroup , or else it has the form:where
is the smallest positive integer in .
has a least element.Division Algorithm
For any integer and positive integer , there are integers 
and s.t. 
and 
and positive integer , there are integers and s.t. and 
?Let and be integers, not both zero, and let 
be their greatest common divisor, the positive integer that generates the subgroup 
, i.e. 
. Then
and be integers, not both zero, and let be their greatest common divisor, the positive integer that generates the subgroup , i.e. . Then- divides and .
- If an integer
divides both and , it also divides - There are integers and
such that 
Fundamental Theorem of Arithmetic
If 
is a prime integer and divides , then divides or divides .
is a prime integer and divides , then divides or divides .Least Common Multiple
If 
and 
, where 
, then is the least common multiple of 
such that
and , where , then is the least common multiple of such that- and divide
- If divide
, then 
divide -
. The cyclic group generated by 
Properties of Cyclic Subgroups
Let 
be the cyclic subgroup of generated by 
and let set 
. Then
be the cyclic subgroup of generated by and let set . Then -
-
- Suppose
. Then 
for some 
and 
are distinct elements of .
. Then 
for 
be a regular 
is
Homomorphisms
Let 
be two groups and 
a map of the underlying sets. Then 
is a homomorphism if
be two groups and a map of the underlying sets. Then is a homomorphism ifProperties of Group Homomorphisms
Let be a group homomorphism. Then
be a group homomorphism. Then- If
are elements of , then 
-
-
and 
Composition of homomorphisms
If 
and 
are group homomorphisms, then
is a group homomorphism.
and are group homomorphisms, then is a group homomorphism.Permutation Matricies
Let 
, 
be the standard basis vectors for 
. Then the map from 
to its permutation matrix is a homomorphism defined as
, be the standard basis vectors for . Then the map from to its permutation matrix is a homomorphism defined asSign Homomorphism
Let the permutation matrix homomorphism. Then 
is the sign homomorphism.
the permutation matrix homomorphism. Then is the sign homomorphism.- The image of the sign =
- The kernel of the sign is called the alternating group
, a \in G. Then
, and 
. TFAE:
is called normal if 
, 
implies 
Property (2) of the Kernel
The kernel of a homomorphism is a normal subgroup
Also, every normal subgroup is the kernel of some homomorphism
Also, every normal subgroup is the kernel of some homomorphism
Inverse of Isomorphism
If is an isomorphism, then the inverse map 
is also an isomorphism
is an isomorphism, then the inverse map is also an isomorphismPartition
A partition 
of a set is a subdivision of into nonoverlapping, nonempty subsets:
union of disjoint nonempty subsets
of a set is a subdivision of into nonoverlapping, nonempty subsets: union of disjoint nonempty subsetsEquivalence Relation
A relation that holds between certain pairs of elements of . We may write it as 
. An equivalence relation is required to be:
. We may write it as . An equivalence relation is required to be:- Transitive: If
and 
, then 
- Symmetric: If then
- Reflexive: For all ,
Equivalence Relation and Partition
An equivalence relation on a set determines a partition of , and conversely.
determines a partition of , and conversely.Equivalence Classes
Given an equivalence relation on a set , the subsets of that are equivalence classes partition
, the subsets of that are equivalence classes partition Coset Partitions
The left Cosets of a subgroup 
partition . Similarly for right Cosets
partition . Similarly for right Cosets
Kartensatzinfo:
Autor: CoboCards-User
Oberthema: Mathematics
Thema: Abstract Algebra
Schule / Uni: Rice University
Ort: Houston
Veröffentlicht: 12.02.2017
Tags: Ronan Mukamel
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