Abstract Algebra

Front 1

Subset of a set

Back 1

Every element of B is in A

B ⊆ A

Front 2

Proper/Improper Subset

Back 2

A is the improper subset of A; any other subset B is a proper sunset, B ⊂ A

Front 3

Cartesian Product

Back 3

A × B = { (a, b) | a ∈ A and b ∈ B }

Front 4

Relation

Back 4

A relation between A and B is a subset /R/ of A × B

Front 5

Equality Relation

Back 5

Subset { (a, a) | a ∈ A } of A × A

Front 6

Function/Mapping

Back 6

A relation /f/ between X and Y with the property that each x ∈ X appears exactly once in /f/

Front 7

Domain/Codomain/Range

Back 7

For any function between sets X and Y, the domain is X, the codomain is Y, and the range is { f(x) | x ∈ X }

Front 8

Cardinality

Back 8

|X| is the number of elements in X

Front 9

One to one/Onto

Back 9

A function is one to one if f(x1) = f(x2) only when x1 = x2. The function is onto Y if the range of /f/ is Y

Front 10

Partition/Disjoint/Cells

Back 10

Sets are disjoint if no two of them have any elements in common. A partition of a set S is a collection of subsets of S such that every s ∈ S is in exactly one of the subsets. The subsets are called cells of S

Front 11

Residue classes modulo n

Back 11

A partition of Z/+ according to whether the remainder is 0, 1, 2 ... n-1 when a positive integer is divided by n.

Front 12

Equivalence Relation

Back 12

An equivalence relation /R/ on a set S exists if it satisfies, for all x, y and z ∈ S

x /R/ x (reflexive)

If x /R/ y then y /R/ x (symmetric)

If x /R/ y and y /R/ z, then x /R/ z (transitive)

Front 13

Equivalence Class

Back 13

Each cell in the partition arising from an equivalence relation

Front 14

Nth Roots of Unity

Back 14

U sub n = { z ∈ /C | z^n = 1 }

Front 15

Binary Operation

Back 15

A function * mapping S × S into S. For each (a, b) ∈ S × S, *((a, b)) is denoted as a * b.

Front 16

Closed under *, Induced Operation

Back 16

A subset H is closed under * if for all a, b ∈ H, a * b ∈ H. The binary operation given by restricting * to H is the induced operation of * on H.

Front 17

Commutativity

Back 17

A binary operation is commutative iff a * b = b * a for all a, b ∈ S

Front 18

Associativity

Back 18

A binary operation * is associative if (a * b) * c = a * (b * c) for all a, b, c ∈ S

Front 19

Reflexive

Back 19

A relation is reflexive if x R x for all x ∈ S

Front 20

Symmetric

Back 20

A relation is symmetric if x R y implies y R x for all x, y ∈ S

Front 21

Transitive

Back 21

A relation is transitive when, if x R y and y R z, then x R z, for all x, y, z ∈ S

Front 22

Idempotent

Back 22

The idempotents of a binary operation * are all elements a ∈ S | a * a = a

Front 23

Binary algebraic structure

Back 23

A binary algebraic structure (S, *) is a set S together with a binary operation * on S

Front 24

Isomorphism

Back 24

An isomorphism is a one-to-one function f mapping S onto S’ such that f(x * y) = f(x) * f(y) for all x, y ∈ S

Front 25

One-to-one Correspondence

Back 25

A mapping between sets which demonstrates that both sets have equal cardinality

Front 26

Identity Element

Back 26

An element e of S is an identity element if e * s = s * e = s for all s ∈ S

Front 27

Group

Back 27

A group (G, *) is a set G closed under a binary operation * such that:

1. * is associative for all a, b ∈ G

2. There is an identity element e such that e * x = x * e = x for all x ∈ G

3. For every element a ∈ G there is an inverse a’ ∈ G such that a * a’ = a’ * a = e

Front 28

Abelian

Back 28

A group (G, *) is abelian if its binary operation * is commutative

Front 29

General Linear Group of degree n

Back 29

GL(n, /R) is the group of invertible n × n matrices with real coefficients

Front 30

Left and Right Cancellation

Back 30

LC: a * b = a * c —> b = c

And

RC: b * a = c * a —> b = c

Follows from three main properties of groups: associativity, inverses, and an identity.

Front 31

Semigroup

Back 31

A set S with a binary operation * that is associative

Front 32

Monoid

Back 32

A set M with a binary operation * such that:

1. * is associative

2. There is an identity element e such that e * m = m * e = m for all m ∈ M

Front 33

Order

Back 33

The order of a group |G| is the number of elements in G. The order of an element a ∈ G is the number of elements in the cyclic subgroup generated by a, ||

Front 34

Subgroup

Back 34

If a subset H of a group G is closed under the binary operation of G, and if H with the induced operation is itself a group, then H is a subgroup of G and H <= G

Front 35

Proper, Improper, Trivial Subgroups

Back 35

If G is a group, then G itself is the improper subgroup of G, and all other subgroups are proper subgroups. All groups contain the trivial subgroup {e}, and all other subgroups are nontrivial

Front 36

Cyclic Subgroup

Back 36

The subgroup {a^n | n ∈ /Z} for some element a ∈ G is the cyclic subgroup of G generated by a written

Front 37

Cyclic Group/Generator

Back 37

An element a of a group G generates G and is a generator for G if = G. A group is cyclic if there is some element in a ∈ G that generates G.

Front 38

Greatest Common Divisor

Back 38

For any two positive integers r and s, the positive generator d of the cyclic group:

{nr + ms | n,m ∈ /Z}

is the greatest common divisor of r and s.

d = gcd(r, s)

Front 39

Relative Prime

Back 39

Two integers are relatively prime if their gcd is 1

Front 40

Permutation of a Set

Back 40

A function φ : A —> A that is one-to-one and onto

Front 41

Symmetric Group on n Letters

Back 41

For any finite set A with elements {1, 2, ... n}, the group of permutations of A is the symmetric group on n letters Sn

Front 42

Nth Dihedral Group

Back 42

The nth dihedral group Dn is the group of symmetries of a regular n-gon

Front 43

Image of H under f

Back 43

For any function f : A —> B and any subset H of A, the image of H under f is { f(h) | h ∈ H } = f[H]

Front 44

Left and Right Regular Representation of a Group

Back 44

The left regular representation of a group G is the map φ: G —> SG where φ(x) = λx = xg for all g ∈ G

The right regular representation of G is the map μ: G —> SG where μ(x) = ρx^-1 = gx’ for all g ∈ G

Front 45

Cycle

Back 45

A permutation σ ∈ Sn is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit.

Front 46

Left and Right Cosets

Back 46

Let H be a subgroup of the group G. The subset aH = {ah | h ∈ H} is the left coset of H containing a, while the subset Ha = {ha | h ∈ H} is the right coset of H containing a.

Front 47

Index

Back 47

Let H be a subgroup of the group G. The index (G : H) of H in G is the number of left cosets of H in G.