TheoremsDefinitionsArticles

Catalog of Definitions. Used Fraleigh's "A First Course in Abstract Algebra" as reference.

Definition 1. Group
A group is a set G together with a binary operation ∗ such that:1. (Closure) ∀a,b∈G,a∗b∈G2. (Associativity) ∀a,b,c∈G,(a∗b)∗c=a∗(b∗c)3. (Identity) ∃e∈G such that ∀a∈G,e∗a=a∗e=a4. (Inverse) ∀a∈G,∃a−1∈G such that a∗a−1=a−1∗a=e,\text{A group is a set } G \text{ together with a binary operation } * \text{ such that:} \\ \quad 1. \text{ (Closure) } \forall a, b \in G, a * b \in G \\ \quad 2. \text{ (Associativity) } \forall a, b, c \in G, (a * b) * c = a * (b * c) \\ \quad 3. \text{ (Identity) } \exists e \in G \text{ such that } \forall a \in G, e * a = a * e = a \\ \quad 4. \text{ (Inverse) } \forall a \in G, \exists a^{-1} \in G \text{ such that } a * a^{-1} = a^{-1} * a = e,
Definition 2. Subgroup
Let G be a group. A subset H⊆G is a subgroup if:1. The identity e∈H2. If a,b∈H, then ab∈H3. If a∈H, then a−1∈H\text{Let } G \text{ be a group. A subset } H \subseteq G \text{ is a subgroup if:} \\ \quad 1. \text{ The identity } e \in H \\ \quad 2. \text{ If } a, b \in H, \text{ then } ab \in H \\ \quad 3. \text{ If } a \in H, \text{ then } a^{-1} \in H
Definition 4. Homomorphism
Let G and K be two groups.A map ϕ:G→K is called a homomorphismif ∀a,b∈G,ϕ(ab)=ϕ(a)ϕ(b) To be homomorphic haiku: operate then mapmust give the same result asmap then operate\text{Let } G \text{ and } K \text{ be two groups.} \\ \text{A map } \phi: G \to K \text{ is called a homomorphism} \\ \text{if } \forall a, b \in G, \phi(ab) = \phi(a)\phi(b) \\ \ \\ \text{To be homomorphic haiku:} \\ \ \\ \text{operate then map} \\ \text{must give the same result as} \\ \text{map then operate}
Definition 5. Isomorphism
An isomorphism is a bijective homomorphism.Written as G≅K, where G and K are groups.\text{An isomorphism is a bijective homomorphism.} \\ \text{Written as } G \cong K, \text{ where } G \text{ and } K \text{ are groups}.
Definition 6. Kernel of a Homomorphism
Let ϕ:G→K be a homomorphism between groups G and K.The kernel of ϕ is the set:ker⁡(ϕ)={x∈G ∣ ϕ(x)=eK}\text{Let } \phi: G \to K \text{ be a homomorphism between groups } G \text{ and } K. \\ \text{The kernel of } \phi \text{ is the set:} \\ \ker(\phi) = \{ x \in G \ | \ \phi(x) = e_K \}
Definition 7. Normal Subgroup
A subgroup H of a group G is called normal (H⊴G)if ∀g∈G,∀h∈H,ghg−1∈H. H⊴G if H⊆G.H◃G if H⊂G.\text{A subgroup } H \text{ of a group } G \text{ is called normal }( H \trianglelefteq G ) \\ \text{if } \forall g \in G, \forall h \in H, ghg^{-1} \in H. \\ \ \\ H \trianglelefteq G \text{ if } H \subseteq G. \\ H \triangleleft G \text{ if } H \subset G.
Definition 8. Generating Set
Let G be a group and let S⊆G be nonempty.Denote S−1={a−1∣a∈S}.Then ⟨S⟩={x1x2...xn ∣ n∈N,xi∈S∪S−1∪{e}}. ⟨S⟩ is created by choosing different n’sand finding the product of 1 through n elements of S∪S−1∪{e}. For a single element:⟨g⟩={gn ∣ n∈Z}\text{Let } G \text{ be a group and let } S \subseteq G \text{ be nonempty.} \\ \text{Denote } S^{-1} = \{ a^{-1} | a \in S \}. \\ \text{Then } \langle S \rangle = \{x_1x_2 ... x_n \ | \ n \in \N, x_i \in S \cup S^{-1} \cup \{e\}\}. \\ \ \\ \langle S \rangle \text{ is created by choosing different } n \text{'s} \\ \text{and finding the product of 1 through n elements of } S \cup S^{-1} \cup \{e\}. \\ \ \\ \text{For a single element:} \\ \langle g \rangle = \{g^n \ | \ n \in \Z \}
Definition 9. Finitely Generated Group
A group is finitely generated ifthere exists a finite nonempty subset of Ss.t. G is generated by S.In other words, G=⟨S⟩.\text{A group is finitely generated if} \\ \text{there exists a finite nonempty subset of } S \\ \text{s.t. } G \text{ is generated by } S. \\ \text{In other words, } G = \langle S \rangle.
Definition 10. Cyclic Group
A group G is cyclic if the group is generated by a single element. i.e. if G=⟨a⟩ for some a∈G.\text{A group } G \text{ is cyclic if } \\ \text{the group is generated by a single element. } \\ \text{i.e. if } G = \langle a \rangle \text{ for some } a \in G.
Definition 11. Relation
Let X be a set.A relation on X is a subset of X×X. In other words, a relation is an ordered pair. For example, the point (1,2) relates 1 and 2 in R2.You would write 1∼2 in this case.\text{Let } X \text{ be a set.} \\ \text{A relation on X is a subset of } X \times X. \\ \ \\ \text{In other words, a relation is an ordered pair}. \\ \ \\ \text{For example, the point } (1, 2) \text{ relates 1 and 2 in } \R^2. \\ \text{You would write } 1 \sim 2 \text{ in this case.}
Definition 12. Properties of Relations
A relation on X is said to be:1◯ reflexive: if x∼x  ∀x∈X2◯ symmetric: if x∼y  ⟹  y∼x  ∀x,y∈X.3◯ transitive: if x∼y∧y∼z  ⟹  x∼z     ∀x,y,z∈X4◯ anti-symmetric: if x∼y∧y∼x  ⟹  x=y     ∀x,y∈X.\def\circled#1{\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {#1}}}} \text{A relation on } X \text{ is said to be:} \\ \circled{1} \text{ reflexive: if } x \sim x \ \ \forall x \in X \\ \circled{2} \text{ symmetric: if } x \sim y \implies y \sim x \ \ \forall x,y \in X. \\ \circled{3} \text{ transitive: if } x \sim y \wedge y \sim z \implies x \sim z \\ \ \ \ \ \ \forall x, y, z \in X \\ \circled{4} \text{ anti-symmetric: if } x \sim y \wedge y \sim x \implies x = y \\ \ \ \ \ \ \forall x, y \in X.
Definition 13. Order Relation
An order relation (or partial order) on Xis a relation that is reflexive, antisymmetric, and transitive. a≤b and a ∣ b (a "divides" b)are examples of order relations.\text{An order relation (or partial order) on } X \\ \text{is a relation that is reflexive, antisymmetric, and transitive.} \\ \ \\ a \leq b \text{ and } a \ | \ b \text{ (a "divides" b)} \\ \text{are examples of order relations.}
Definition 14. Equivalence Relation
An equivalence relation on Xis a relation that is reflexive, symmetric, and transitive. For example, say two M&Ms are the same if they have the same colorThis is denoted as a∼b  ⟺  (a has same color as b). Considering different equivalence relations on a set gives different partitions.So a single set can be observed in many different ways.\text{An equivalence relation on } X \\ \text{is a relation that is reflexive, symmetric, and transitive.} \\ \ \\ \text{For example, say two M} \& \text{Ms are the same if they have the same color} \\ \text{This is denoted as } a \sim b \iff (a \text{ has same color as } b). \\ \ \\ \text{Considering different equivalence relations on a set gives different partitions.} \\ \text{So a single set can be observed in many different ways.}
Definition 15. Equivalence Class
Given an equivalence relation ∼ on a set X,the equivalence class of x∈X is the set:[x]={y∈X ∣ x∼y} This was presented as a theorem:1)  ∀x,y∈X  if x∼y then [x]=[y]  if x≁y then [x]∩[y]=∅.2)  ⋃x∈X[x]=X.\text{Given an equivalence relation } \sim \text{ on a set } X, \\ \text{the equivalence class of } x \in X \text{ is the set:} \\ [x] = \{ y \in X \ | \ x \sim y \} \\ \ \\ \text{This was presented as a theorem:} \\ \quad 1) \ \ \forall x, y \in X \\ \quad \quad \ \ \text{if } x \sim y \text{ then } [x] = [y] \\ \quad \quad \ \ \text{if } x \not \sim y \text{ then } [x] \cap [y] = \empty. \\ \quad 2) \ \ \bigcup_{x \in X} [x] = X.
Definition 16. Left Coset
Let H≤G and x∈G.The set {xh ∣ h∈H} is called the left coset by x.Written as xH.\text{Let } H \leq G \text{ and } x \in G. \\ \text{The set } \{ xh \ | \ h \in H \} \text{ is called the left coset by } x. \\ \text{Written as } xH.
Definition 17. Automorphism
An automorphism is an isomorphism that maps from G to G.Aut(G) is the set of all automorphisms of G.\text{An automorphism is an isomorphism that maps from } G \text{ to } G. \\ \text{Aut}(G) \text{ is the set of all automorphisms of G.}
Definition 18. Inner Automorphism
An inner automorphism is an automorphism of the form ϕg:G→G, defined by ϕg(x)=gxg−1  for x,g∈G. We denote Inn(G)={ϕg ∣ g∈G}as the set of all inner automorphisms of G.\text{An inner automorphism is an automorphism } \\ \text{of the form } \phi_g: G \to G, \text{ defined by } \phi_g(x) = gxg^{-1} \ \text{ for } x, g \in G. \\ \ \\ \text{We denote }\text{Inn}(G) = \{\phi_g \ | \ g \in G \} \\ \text{as the set of all inner automorphisms of } G.
Definition 19. Symmetric Group
Let X be a set. Let SX={f:X→X ∣ f is bijective}.SX with composition is called the symmetric group. For n∈N and X={1,2,...,n},We denote Sn for SX. For example S2 is the group of bijective functions over X={1,2},and only has two elements: the identity function and a single transposition.In other words the only actions you can do in S2 is stay in place or swap places.\text{Let } X \text{ be a set. Let } S_X = \{f: X \to X \ | \ f \text{ is bijective} \}. \\ S_X \text{ with composition is called the symmetric group.} \\ \ \\ \text{For } n \in \N \text{ and } X = \{ 1, 2, ..., n \}, \\ \text{We denote } S_n \text{ for } S_X. \\ \ \\ \text{For example } S_2 \text{ is the group of bijective functions over } \\ X = \{ 1, 2 \}, \text{and only has two elements: the identity function and a single transposition}. \\ \text{In other words the only actions you can do in } S_2 \text{ is stay in place or swap places.}
Definition 20. Permutation
A permutation is a bijective mapping from a set onto itself\text{A permutation is a bijective mapping from a set onto itself}
Definition 21. Transposition
Let n≥2.A transposition is a permutation τ∈Snof the form (i j) for 1≤i≠j≤n.\text{Let } n \geq 2. \\ \text{A transposition is a permutation } \tau \in S_n \\ \text{of the form } (i \ j) \text{ for } 1 \leq i \neq j \leq n.
Definition 22. Even or Odd Permutation
A permutation is called even ifit is a product of an even number of transpositions. A permutation is called odd ifit is a product of an odd number of transpositions.\text{A permutation is called even if} \\ \text{it is a product of an even number of transpositions.} \\ \ \\ \text{A permutation is called odd if} \\ \text{it is a product of an odd number of transpositions.}
Definition 23. Index
Let H≤G.The index of H in G isthe cardinality of the coset space ∣G/H∣.Denoted by [G:H].\text{Let } H \leq G. \\ \text{The index of H in G is} \\ \text{the cardinality of the coset space } |G/H|. \\ \text{Denoted by } [G:H].
Definition 24. Direct Product of Groups
Let G1,G2 be groups with g1∈G1 and g2∈G2.The direct product G1×G2is the group of ordered pairs (g1,g2)with the binary operation (g1,g2)⋅(h1,h2)=(g1h1,g2h2)called the cartesian product For example the point (1,2)is an element in R2and R2=R×R.\text{Let } G_1, G_2 \text{ be groups with } g_1 \in G_1 \text{ and } g_2 \in G_2. \\ \text{The direct product } G_1 \times G_2 \\ \text{is the group of ordered pairs } (g_1, g_2) \\ \text{with the binary operation } \\ (g_1, g_2) \cdot (h_1, h_2) = (g_1h_1, g_2h_2) \\ \text{called the cartesian product} \\ \ \\ \text{For example the point } (1, 2) \\ \text{is an element in } \R^2 \\ \text{and } \R^2 = \R \times \R.
Definition 25. Ring
A ring is a set Requipped with two associative binary operations+:R×R→R,⋅:R×R→R satisfying:  1) (R,+) is an abelian group  2)  ∀a,b,c∈R,      (a+b)⋅c=(a⋅c)+(b⋅c)      a⋅(b+c)=(a⋅b)+(a⋅c).\text{A ring is a set } R \\ \text{equipped with two associative binary operations} \\ +: R \times R \to R, \\ \cdot: R \times R \to R \\ \text{ satisfying:} \\ \ \\ \ 1) \ (R, +) \text{ is an abelian group} \\ \ \\ \ 2) \ \ \forall a, b, c \in R, \\ \ \ \ \ \ \ (a + b) \cdot c = (a \cdot c) + (b \cdot c) \\ \ \ \ \ \ \ a \cdot (b + c) = (a \cdot b) + (a \cdot c).
Definition 26. Commutative Ring
A ring R is commutative ifxy=yx  ∀x,y∈R.\text{A ring } R \text{ is commutative if} \\ x y = y x \ \ \forall x,y \in \R.
Definition 27. Unital Ring
Put simply, a ring is unital if it has both neutral elements. Let (R,+,⋅) be a ring.We say R is unital if it hasa neutral element for the operation ⋅. We denote the neutral element of (R,+) with 0and the neutral element of (R,⋅) with 1\text{Put simply, a ring is unital if it has both neutral elements.} \\ \ \\ \text{Let } (R,+,\cdot) \text{ be a ring.} \\ \text{We say } R \text{ is unital if it has} \\ \text{a neutral element for the operation } \cdot. \\ \ \\ \text{We denote the neutral element of } (R, +) \text{ with } 0 \\ \text{and the neutral element of } (R, \cdot) \text{ with } 1
Definition 28. Zero Divisor
A non-zero element x∈R is a zero divisor ifthere exists another non-zero element y∈Rsuch that xy=0. In non-commutative rings this is called the left zero-divisor.In commutative rings left and right zero-divisors are the same.\text{A non-zero element } x \in R \text{ is a zero divisor if} \\ \text{there exists another non-zero element } y \in R \\ \text{such that } xy = 0. \\ \ \\ \text{In non-commutative rings this is called the left zero-divisor.} \\ \text{In commutative rings left and right zero-divisors are the same.}
Definition 29. Integral Domain
A Ring R is an integral domain ifR is a untial commutative ring andR has no zero divisors.\text{A Ring } R \text{ is an integral domain if} \\ R \text{ is a untial commutative ring and} \\ R \text{ has no zero divisors}.
Definition 30. Division Ring
R is a division ring ifR is a unital ring and(R−{0},⋅) is a group. This is if every non-zero elementhas a multiplicative inverse for this operationR \text{ is a division ring if} \\ R \text{ is a unital ring and} \\ (R-\{0\}, \cdot) \text{ is a group.} \\ \ \\ \text{This is if every non-zero element} \\ \text{has a multiplicative inverse for this operation}
Definition 31. Field
A commutative division ring is called a field.\text{A commutative division ring is called a field.}
Definition 32. Subring
Let (R,+,⋅) be a ringA subset S⊆R is called a subring if:1)  (S,+) is a subgroup of (R,+)2)  ∀x,y∈S, we have x⋅y∈S.\text{Let } (R, +, \cdot) \text{ be a ring} \\ \text{A subset } S \subseteq R \text{ is called a subring if:} \\ 1) \ \ (S, +) \text{ is a subgroup of } (R, +) \\ 2) \ \ \forall x, y \in S, \text{ we have } x \cdot y \in S.
Definition 33. Ring Homomorphism
Let R and S be ringsand ϕ:R→S be a map.We say ϕ is a ring homomorphismif ∀a,b∈R.1)  ϕ(a+b)=ϕ(a)+ϕ(b)2)  ϕ(ab)=ϕ(a)ϕ(b)\text{Let } R \text{ and } S \text{ be rings} \\ \text{and } \phi: R \to S \text{ be a map}. \\ \text{We say } \phi \text{ is a ring homomorphism} \\ \text{if } \forall a, b \in R. \\ \quad 1) \ \ \phi (a+b) = \phi(a) + \phi(b) \\ \quad 2) \ \ \phi(ab) = \phi(a)\phi(b)
Definition 34. Kernel of Ring Homomorphism
Let ϕ:R→S be a ring homomorphism.The kernel of ϕ is ker⁡(ϕ)={r∈R ∣ ϕ(r)=0}.\text{Let } \phi: R \to S \text{ be a ring homomorphism}. \\ \text{The kernel of } \phi \text{ is } \ker(\phi) = \{ r \in R \ | \ \phi(r) = 0 \}.
Definition 35. Ideal
Let I≤R.I is an ideal in Rif ∀a∈R and x∈I,we have ax∈I and xa∈I.In this case we write I⊴R. An ideal I◃R is proper if I≠R.\text{Let } I \leq R. \\ I \text{ is an ideal in } R \\ \text{if } \forall a \in R \text{ and } x \in I, \\ \text{we have } ax \in I \text{ and } xa \in I. \\ \text{In this case we write } I \trianglelefteq R. \\ \ \\ \text{An ideal } I \triangleleft R \text{ is proper if } I \neq R.
Definition 36. Maximal Ideal
I◃R is maximal if I≠R,and if J◃R s.t. I⊆J⊆R,then either J=I or J=R.I \triangleleft R \text{ is maximal if } I \neq R, \\ \text{and if } J \triangleleft R \text{ s.t. } I \subseteq J \subseteq R, \\ \text{then either } J = I \text{ or } J = R.
Definition 37. Polynomial Ring
The polynomial ring R[x]has polynomial elements of the formp(x)=anxn+an−1xn−1+...+a1x+a0with a0,a1,...,an∈R and an≠0.\text{The polynomial ring } \R[x] \\ \text{has polynomial elements of the form} \\ p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0 \\ \text{with } a_0, a_1, ..., a_n \in \R \text{ and } a_n \neq 0.
Definition 38. Division Algorithm
Given polynomials, p(x) and q(x)there are polynomials a(x),r(x)s.t. q(x)=p(x)a(x)+r(x).Where either r(x)=0 or deg⁡(r(x))≤deg⁡(p(x)).\text{Given polynomials, } p(x) \text{ and } q(x) \\ \text{there are polynomials } a(x), r(x) \\ \text{s.t. } q(x) = p(x) a(x) + r(x). \\ \text{Where either } r(x) = 0 \text{ or } \deg(r(x)) \leq \deg(p(x)).
Definition 39. Principal Ideal
A principal ideal ⟨x⟩ is an ideal in a commutative ring R generated by x∈Rwith ⟨x⟩={rx ∣ r∈R}.\text{A principal ideal } \langle x \rangle \text{ is an ideal in } \\ \text{a commutative ring } R \text{ generated by } x \in R \\ \text{with } \langle x \rangle = \{rx \ | \ r \in R \}.
Definition 40. Principal Ideal Domain (PID)
A PID is an integral domain Rs.t. every ideal of R is principal.\text{A PID is an integral domain } R \\ \text{s.t. every ideal of } R \text{ is principal}.
Definition 41. Divides Notation
Let R be a ring and 0≠a,x∈R.We write a∣xiff  ∃b∈R s.t. x=ab.\text{Let } R \text{ be a ring and } 0 \neq a, x \in R. \\ \text{We write } a | x \\ \text{iff } \ \exists b \in R \text{ s.t. } x = ab.
Definition 42. Prime Element
Let R be a commutative ring.An element p∈R is said to be prime ifit is not the zero element or the one elementand whenever p∣ab ∀a,b∈R, then p∣a or p∣b.\text{Let } R \text{ be a commutative ring.} \\ \text{An element } p \in R \text{ is said to be prime if} \\ \text{it is not the zero element or the one element} \\ \text{and whenever } p | ab \ \forall a, b \in R, \text{ then } p | a \text{ or } p | b.
Definition 43. Irreducible/Prime Element
Let R be an ID.A non-zero and non-invertible (not one) element x∈Ris called prime iff whenever a,b∈R and x=ab,then either a is invertible or b is invertible. In Unique Factor Domains irreducable and prime is the same.So for our purposes treat them as having both properties.\text{Let R be an ID.} \\ \text{A non-zero and non-invertible (not one) element } x \in R \\ \text{is called prime iff whenever } a, b \in R \text{ and } x = ab, \\ \text{then either } a \text{ is invertible or } b \text{ is invertible.} \\ \ \\ \text{In Unique Factor Domains irreducable and prime is the same.} \\ \text{So for our purposes treat them as having both properties.}
Definition 44. Prime Ideal
Let R be an ID.A proper ideal I◃R is called primeiff whenever ab∈R and ab∈Ithen either a∈I or b∈I.\text{Let } R \text{ be an ID.} \\ \text{A proper ideal } I \triangleleft R \text{ is called prime} \\ \text{iff whenever } ab \in R \text{ and } ab \in I \\ \text{then } \text{either } a \in I \text{ or } b \in I.