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

(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.

Symmetric Group

The group of permutations of the set of indices is called the symmetric group and is denoted by .

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 .

where is the smallest positive integer in .

Division Algorithm

For any integer 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

- 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 .

Least Common Multiple

If and , where , then is the least common multiple of such that

- and divide
- If divide , then divide

Properties of Cyclic Subgroups

Let be the cyclic subgroup of generated by and let set . Then

- Suppose . Then for some and are distinct elements of .

Homomorphisms

Let be two groups and a map of the underlying sets. Then is a homomorphism if

Properties of Group Homomorphisms

Let be a group homomorphism. Then

- If are elements of , then

Composition of homomorphisms

If and are group homomorphisms, then

is a group homomorphism.

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

Sign Homomorphism

Let the permutation matrix homomorphism. Then is the sign homomorphism.

- The image of the sign =
- The kernel of the sign is called the alternating group

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

Partition

A partition of a set is a subdivision of into nonoverlapping, nonempty subsets:

union of disjoint nonempty subsets

union of disjoint nonempty subsets

Equivalence Relation

A relation that holds between certain pairs of elements of . 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.

Equivalence Classes

Given an equivalence relation on a set , the subsets of that are equivalence classes partition

Coset Partitions

The left Cosets of a subgroup 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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