Abstract Algebra Definitions
Definition: Ring
A nonempty set 2 Closed Binary Operations 1st Operation- Abelian Group Associative, Commutative, Identity, Every element-inverse Associative under multiplication, Contains a multiplicative inverse Distributive with respect to addition
Definition: Commutative Ring
A ring is commutative if it is abelian to the second operation.
Group: SL2(R)
Special Linear Group 2x2 invertible matrices det A=1 Nonabelian
Definition: Zero Divisor
2 elements that multiply to the multiplicative identity.
Definition: Integral Domain
A Commutative Ring A Ring with a Unique Multiplicative Identity No Zero Divisors
Definition: Field
A Ring that is an Integral Domain A ring that is a Division Ring Has a Unity
Definition: Permutation
A bijection from a set to itself.
Definition: Extension Field
A field E is an extension field of F if F is a subfield of E. The field F is called the base field.
Definition: Cyclic Group
A group that can be generated by a single element There exists an element a in the group G such that, <a>=G
Definition: Abelian Group
A group that is commutative.
Definition: Homomorphism
A homomorphism between groups (G,+) and (H,*) such that φ:G-> H, φ(g1+g2)=φ(g1)*φ(g2)
Definition: Isomorphism
A mapping from one group to another that preserves operation and is a bijection.
Definition: Unity
A multiplicative Identity
Definition: Transcendental Number
A number that is not algebraic
Definition: Group
A set of objects that must be: Closed under operation, Contain an identity element Each element has an inverse Associative
Definition: Subgroup
A subdivision of a group. Has an identity Has an inverse The operation is closed
Definition: Normal Subgroup
A subgroup H of a group G if gH=Hg for all g in G.
Definition: Ideal
An ideal in a ring R is a subring I of R such that if a is in I and r is in R, then both ar and ra are in I; that is rI ⊂ I and Ir ⊂ I for all r in R.
Definition: Algebraic Number
Any element α in an extension field E over F is algebraic over F if f(α)=0 for some nonzero polynomial f(x) in F[x]. An algebraic number is one that has a polynomial which has that as a root.
Group: C*
Complex Numbers excluding 0 Multiplication Identity: 1
Group: GL2(R)
General Linear Group 2x2 invertible matrices det A n=0 Nonabelian
Definition: Direct Product G x H
If (G,+) and (H,*) are groups, and g1,g2 in G and h1,h2 in H. Then the direct product of G x H. (g1,h1)(g2,h2)= (g1+g2 , h1*h2) The operation is within the groups.
Definition: Factor Group G/N
If N is a normal subgroup of a group G, then the cosets of N in G form a group G/N, under the operation (aN)(bN)=abN
Definition: Kernel of a Homomorphism
If φ is a homomorphism φ: G -> H, the kernel of φ is the set {g∈G | φ(g) = e_{H}}.
Group: Zn
Integers mod n Addition Mod n, Abelian Identity: 0
Definition: Left Coset
Let G be a group and H be a subgroup of G. For g in G, gH={gh:h in H}.
Definition: Even/Odd Permutation
Odd- A bijection from a set to itself that has an even number of elements that can be made into a set of odd transpositions. Even- A bijection from a set to itself that has an odd number of elements that can be made into a set of even transpositions.
Group: Q(8)
Quaternians Multiplication Nonabelian Identity: 1 +-1 +-i +-j +-k
Group: R*
Real Numbers excluding 0 Multiplication Identity:1
Group: U(n)
The Natural numbers relatively prime to Z_{n} Multiplication mod n Abelian Identity: 0
Definition: Index of H in G, [G:H]
The amount of distinct left or right cosets of a subgroup H on a group G [G:H]= |G|/|H|
Definition: Order of an Element
The least positive integer n such that a^n = e. (In additive notation, this would be na = 0.) The fewest # of times when operated upon to reach identity.
Definition: Order of a Group
The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G.
Group: Dn
The rigid motions of a regular n-gon. N-vertices Nonabelian Not Cyclic. Operation is composition.
Group: Z
The set of Integers Addition Abelian Cyclic Identity:0
Definition: Factor Ring
The set of all equivalence classes is denoted by R/I, thus: (a+I)+(b+I)=(a+b)+I (a+I)(b+I)=(ab)+I
Group: Sn
The set of permutations on n-objects. Composition Nonabelian Not cyclic Identity: p0
