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## Abstract Algebra Midterm 1 (46 Karten)

Law of Composition
A function of two variables, or a map

Associativity of LOC
A LOC is associative if

ie order of operations does not matter
Commutativity of LOC
LoC is commutative if

Identity element
An element

is an identity element if

Invertible
An element 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
Abelian Group
A group whose law of composition is commutative
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 .
Order of a Group
The order of a group is its number of elements (could be ) denoted
Subgroup
A subgroup of is a subset s.t.

• (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 .
Well-Ordering Principle
Any non-empty subset has a least element.
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
Relatively Prime
and are relatively prim if
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
Cyclic Group
Fix a group and an element . The cyclic group generated by is

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 .
Order of
Let . Then is the order of .
Multiplying elements in
Let be the order of . Then

for
Dihedral Group
Let be a regular -gon. Then the dihedral group of order is

Where,

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
Image of a homomorphism
The image of a homomorphism is

Kernel of a homomorphism
The kernel of a homomorphism is

Property of Kernel and Image
For any homomorphism , and
Composition of homomorphisms
If 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

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
Isomorphisms
A homomorphism is an isomorphism if it is bijective
Left Coset
Let , a \in G. Then

is a left coset of
Properties of Kernel
Let be a group homomorphism, , and . TFAE:

Injectivity of Homomorphisms
A homomorphism is injective if and only if
Normal Subgroups
A subgroup 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
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
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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