TheoremsDefinitionsArticles

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

Definition 1. Group
A group is a set Galong 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{along 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 \\ \quad \quad \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 3. 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 4. 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 5. Kernel of a Homomorphism
Let ϕ:G→K be a homomorphismbetween 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} \\ \text{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 6. 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 7. 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 8. 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 9. 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 10. 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 11. 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 12. 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 13. 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 setgives 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,} \\ \text{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} \\ \text{gives different partitions.} \\ \text{So a single set can be observed in many different ways.}
Definition 14. 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 15. 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 16. 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 17. 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 18. 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 S2is stay in place or swap places. Even more generally,Sn has n! permutations.\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:} \\ \text{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.} \\ \ \\ \text{Even more generally,} \\ S_n \text{ has } n! \text{ permutations.}
Definition 19. 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 20. Cycles
A cycle is a simple way to represent a permutation. Given a permutation σ∈S4,Define σ=(1 2 3 4)This notation means 1→22→33→44→1 σ=(1 2)(3 4) means1→22→13→44→3 σ=(1 2) means1→22→13→34→4\text{A cycle is a simple way to represent a permutation.} \\ \ \\ \text{Given a permutation } \sigma \in S_4, \\ \text{Define } \sigma = (1 \ 2 \ 3 \ 4) \\ \text{This notation means } \\ 1 \to 2 \\ 2 \to 3 \\ 3 \to 4 \\ 4 \to 1 \\ \ \\ \sigma = (1 \ 2) (3 \ 4) \text{ means} \\ 1 \to 2 \\ 2 \to 1 \\ 3 \to 4 \\ 4 \to 3 \\ \ \\ \sigma = (1 \ 2) \text{ means} \\ 1 \to 2 \\ 2 \to 1 \\ 3 \to 3 \\ 4 \to 4
Definition 21. Cycle Multiplication
To multiply cycles you compose the two permutationsστ can be read as apply σ and then apply τ. Let σ,τ∈S4 withσ=(a b c d) and τ=(a d)(b c). To find στ,we evaluate τ∘σ or τ(σ).Can be written as ((a d)(b c))(a b c d). Starting with a the first element in σ,a→σb then b→τcc→σd then d→τaThis means a and c forms a cycle (a c).Now we go to the next element we haven’t evaluated.b→σc then c→τb.So b is in a 1-cycle, because it maps to itself.Finally we evaluate d,d→σa then a→τd.Therefore, our final product is (a c).\text{To multiply cycles you compose the two permutations} \\ \sigma \tau \text{ can be read as apply } \sigma \text{ and then apply } \tau. \\ \ \\ \text{Let } \sigma, \tau \in S_4 \text{ with} \\ \sigma = (a \ b \ c \ d) \text{ and } \tau = (a \ d)(b \ c). \\ \ \\ \text{To find } \sigma \tau, \\ \text{we evaluate } \tau \circ \sigma \text{ or } \tau(\sigma). \\ \text{Can be written as } ((a \ d)(b \ c))(a \ b \ c \ d). \\ \ \\ \text{Starting with } a \text{ the first element in } \sigma, \\ a \xrightarrow{\sigma} b \text{ then } b \xrightarrow{\tau} c \\ c \xrightarrow{\sigma} d \text{ then } d \xrightarrow{\tau} a \\ \text{This means } a \text{ and } c \text{ forms a cycle } (a \ c). \\ \text{Now we go to the next element we haven't evaluated.} \\ b \xrightarrow{\sigma} c \text{ then } c \xrightarrow{\tau} b. \\ \text{So } b \text{ is in a 1-cycle, because it maps to itself.} \\ \text{Finally we evaluate } d, \\ d \xrightarrow{\sigma} a \text{ then } a \xrightarrow{\tau} d. \\ \text{Therefore, our final product is } \\ (a \ c).
Cycle multiplication diagram
Definition 22. 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 23. 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 24. 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 25. 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 26. 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 27. 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 28. 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 29. 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 ringsleft 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} \\ \text{left and right zero-divisors are the same.}
Definition 30. 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 31. 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 32. Field
A commutative division ring is called a field.\text{A commutative division ring is called a field.}
Definition 33. 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:} \\ \quad 1) \ \ (S, +) \text{ is a subgroup of } (R, +) \\ \quad 2) \ \ \forall x, y \in S, \text{ we have } x \cdot y \in S.
Definition 34. 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 35. 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 36. 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 37. Quotient Ring
R/I is the set of cosets of I with respect to +.Denote R/I={r+I∣r∈R}. For R/I to be a ring I must be an ideal of R.This is equipped with the operations:(x+I)+(y+I)=(x+y)+I and(x+I)⋅(y+I)=(xy)+I.R/I \text{ is the set of cosets of } I \text{ with respect to } +. \\ \text{Denote } R/I = \{ r+I \mid r \in R \}. \\ \ \\ \text{For } R/I \text{ to be a ring } I \text{ must be an ideal of } R. \\ \text{This is equipped with the operations:} \\ (x+I) + (y+I) = (x+y)+I \text{ and} \\ (x+I) \cdot (y+I) = (xy)+I.
Definition 38. 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 39. 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 40. 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 41. Principal Ideal
A principal ideal ⟨x⟩ is an ideal in a commutative ring R generated by x∈Rwith ⟨x⟩={rx ∣ r∈R}. For example, ⟨6⟩ is a principal ideal.Because ⟨6⟩={n⋅6 ∣ n∈Z}\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 \}. \\ \ \\ \text{For example, } \langle 6 \rangle \text{ is a principal ideal.} \\ \text{Because } \langle 6 \rangle = \{n \cdot 6 \ | \ n \in \Z \}
Definition 42. 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 43. Divides Notation
Let R be a ring and 0≠a,x∈R.We write a∣xiff  ∃b∈R s.t. x=ab. 3∣6 becase 3⋅2=6.\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. \\ \ \\ 3 | 6 \text{ becase } 3 \cdot 2 = 6.
Definition 44. Invertible Element
An element a∈R is invertibleif there exists an element b∈Rs.t. ab=1. An element a∈Zn is invertible iffgcd⁡(a,n)=1. If n is primethen every element in Zn is invertible For example,in Z6 both 1 and 5 are invertible.1⋅1≡1(mod6)5⋅5=25≡1(mod6)inverse of 1 is 1 and inverse of 5 is 5.\text{An element } a \in R \text{ is invertible} \\ \text{if there exists an element } b \in R \\ \text{s.t. } ab = 1. \\ \ \\ \text{An element } a \in \Z_n \text{ is invertible iff} \\ \text{} \gcd(a, n) = 1. \\ \ \\ \text{If } n \text{ is prime} \\ \text{then every element in } \Z_n \text{ is invertible} \\ \ \\ \text{For example,} \\ \text{in } \Z_6 \text{ both } 1 \text{ and } 5 \text{ are invertible.} \\ 1 \cdot 1 \equiv 1 \pmod{6} \\ 5 \cdot 5 = 25 \equiv 1 \pmod{6} \\ \text{inverse of 1 is 1 and inverse of 5 is 5.}
Definition 45. 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 46. Irreducible/Prime Element
Let R be an ID.A non-zero and non-invertible (not one) element x∈Ris called irreducible iff whenever a,b∈R and x=ab,then either a is invertible or b is invertible. This is saying thatan irreducible element cannot be factoredas two non-invertible elements. In Unique Factor Domains (UFDs)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 irreducible iff whenever } a, b \in R \text{ and } x = ab, \\ \text{then either } a \text{ is invertible or } b \text{ is invertible.} \\ \ \\ \text{This is saying that} \\ \text{an irreducible element cannot be factored} \\ \text{as two non-invertible elements.} \\ \ \\ \text{In Unique Factor Domains (UFDs)} \\ \text{irreducable and prime is the same.} \\ \text{So for our purposes treat them as having both properties.}
Definition 47. 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. For example,6Z⊴Z is not a prime ideal.By choosing 2,3∈Z, we have 2⋅3∈Z and 2⋅3∈6Z.However, neither 2 or 3 are in 6Z. 7Z is an example of a prime ideal.Since 7 is a prime integer, its only factors are 1 and 7.\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. \\ \ \\ \text{For example,} \\ 6 \Z \trianglelefteq \Z \text{ is not a prime ideal.} \\ \text{By choosing } 2, 3 \in \Z, \text{ we have } 2 \cdot 3 \in \Z \text{ and } 2 \cdot 3 \in 6 \Z. \\ \text{However, neither } 2 \text{ or } 3 \text{ are in } 6\Z. \\ \ \\ 7 \Z \text{ is an example of a prime ideal.} \\ \text{Since 7 is a prime integer, its only factors are 1 and 7.}
Definition 48. Group Action
Let G be a group and let X be a set.An action of G on X isa group homomorphism α:G→SX. For each g∈G and x∈Xwe denote gx=α(g)(x). An example is rotations of a snowflake.\text{Let } G \text{ be a group and let } X \text{ be a set.} \\ \text{An action of } G \text{ on } X \text{ is} \\ \text{a group homomorphism } \alpha: G \to S_X. \\ \ \\ \text{For each } g \in G \text{ and } x \in X \\ \text{we denote } gx = \alpha(g)(x). \\ \ \\ \text{An example is rotations of a snowflake.}
Definition 49. Kernel of Action
The kernel of the action is ker⁡(α)={g∈G ∣ gx=x  ∀x∈X} In other words, the kernel is the set ofelements of G that when acted upon x dont change x. For example, taking the group Z6.and the set Z3. Let n∈Z6 and x∈Z3.Define the action gx≔(n+x)(mod3).ker⁡(α)={0,3}. If n=0, then 0⋅x=(0+x)(mod3)=x.So 0 is in the kernel. If n=1, then 1⋅0=(1+0)(mod3)=1.So 1 is not in the kernel.\text{The kernel of the action is } \\ \ker(\alpha) = \{g \in G \ | \ gx = x \ \ \forall x \in X \} \\ \ \\ \text{In other words, the kernel is the set of} \\ \text{elements of G that when acted upon x dont change x.} \\ \ \\ \text{For example, taking the group } \Z_6. \\ \text{and the set } \Z_3. \text{ Let } n \in \Z_6 \text{ and } x \in \Z_3. \\ \text{Define the action } gx \coloneqq (n+x) \pmod{3}. \\ \ker(\alpha) = \{ 0, 3 \}. \\ \ \\ \text{If } n = 0, \text{ then } 0 \cdot x = (0+x) \pmod{3} = x. \\ \text{So 0 is in the kernel.} \\ \ \\ \text{If } n = 1, \text{ then } 1 \cdot 0 = (1+0) \pmod{3} = 1. \\ \text{So 1 is not in the kernel.}
Definition 50. Faithful Action
An action is faithful if the kernel is trivial. If ∀g∈G where g≠e, ∃x∈X s.t. gx≠x,then the action is faithful. Or in other words, whenever ker⁡={e}. So when an action is faithful,we can tell different elements of G apart. In the previous example of G=Z6 and X=Z3the action of mod 3 addition is not faithful.Because ker⁡(α)={0,3}≠{0}. An example of a faithful action would begx≔(g+x)(mod3)with G=Z3 and X={1}.0⋅1=(0+1)(mod3)=11⋅1=(1+1)(mod3)=22⋅1=(2+1)(mod3)=3\text{An action is faithful if the kernel is trivial.} \\ \ \\ \text{If } \forall g \in G \text{ where } g \neq e, \ \exists x \in X \text{ s.t. } gx \neq x, \\ \text{then the action is faithful.} \\ \ \\ \text{Or in other words, whenever } \ker = \{e\}. \\ \ \\ \text{So when an action is faithful,} \\ \text{we can tell different elements of } G \text{ apart.} \\ \ \\ \text{In the previous example of } G = \Z_6 \text{ and } X = \Z_3 \\ \text{the action of mod 3 addition is not faithful.} \\ \text{Because } \ker(\alpha) = \{ 0, 3 \} \neq \{0\}. \\ \ \\ \text{An example of a faithful action would be} \\ gx \coloneqq (g+x) \pmod{3} \\ \text{with } G = Z_3 \text{ and } X = \{1 \}. \\ 0 \cdot 1 = (0+1) \pmod{3} = 1 \\ 1 \cdot 1 = (1+1) \pmod{3} = 2 \\ 2 \cdot 1 = (2+1) \pmod{3} = 3
Definition 51. Free Action
An action is free if∀g∈G where g≠e, every x∈X satisfies gx≠x. For example, chooseG=GLn(R) and X=Rn.Let A∈G and v∈X.Define Av≔Av (matrix multiplication). This is faithful since every matrixis a unique linear transformation. However, the action is not free becauseany matrix multiplied by the zero vector is the zero vector.\text{An action is free if} \\ \forall g \in G \text{ where } g \neq e, \text{ every } x \in X \text{ satisfies } gx \neq x. \\ \ \\ \text{For example, choose} \\ G = GL_n(\R) \text{ and } X = \R^n. \\ \text{Let } A \in G \text{ and } v \in X. \\ \text{Define } Av \coloneqq Av \text{ (matrix multiplication)}. \\ \ \\ \text{This is faithful since every matrix} \\ \text{is a unique linear transformation.} \\ \ \\ \text{However, the action is not free because} \\ \text{any matrix multiplied by the zero vector is the zero vector}.
Definition 52. G-set
A G-set is a setXalongside a group action of a group G on X.\text{A G-set is a set} X \\ \text{alongside a group action of a group } G \text{ on } X.
Definition 53. G-orbits
The G-orbits of x is the setGx={gx ∣ g∈G}. Can be thought of as the set of positionsyou can move x to by applying group elements.\text{The G-orbits of } x \text{ is the set} \\ Gx = \{ gx \ | \ g \in G \}. \\ \ \\ \text{Can be thought of as the set of positions} \\ \text{you can move } x \text{ to by applying group elements.}
Definition 54. Stabilizer
The stabilizer of x is the setGx={g∈G ∣ gx=x} The set of elements of Gthat don’t move our given x. The kernel is the set of elements of Gthat don’t move any x.So ker⁡⊆Gx.\text{The stabilizer of } x \text{ is the set} \\ G_x = \{ g \in G \ | \ gx = x \} \\ \ \\ \text{The set of elements of G} \\ \text{that don't move our given } x. \\ \ \\ \text{The kernel is the set of elements of } G \\ \text{that don't move any } x. \\ \text{So } \ker \subseteq G_x.
Definition 55. Fixed Points
Given a G-set X, the set of fixed points isdenoted as XG={x∈X ∣ gx=x  ∀g∈G}. For example, letG=S3 and X=R[x1,x2,x3].For σ∈S3 and p(x1,x2,x3)∈R[x1,x2,x3]Define σp≔p(xσ(1),xσ(2),xσ(3)).So XS3={p∈R[x1,x2,x3] ∣ σp=p  ∀σ∈S3}.XS3=R[e1,e2,e3].R[e1,e2,e3] is the set of symmetric polynomials.Look up the fundamental theorem of symmetric polynomials.\text{Given a G-set } X, \text{ the set of fixed points is} \\ \text{denoted as } X^G = \{x \in X \ | \ gx = x \ \ \forall g \in G \}. \\ \ \\ \text{For example, let} \\ G = S_3 \text{ and } X = \R[x_1, x_2, x_3]. \\ \text{For } \sigma \in S_3 \text{ and } p(x_1, x_2, x_3) \in \R[x_1, x_2, x_3] \\ \text{Define } \sigma p \coloneqq p(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}). \\ \text{So } X^{S_3} = \{p \in \R[x_1, x_2, x_3] \ | \ \sigma p = p \ \ \forall \sigma \in S_3 \}. \\ X^{S_3} = \R[e_1, e_2, e_3]. \\ \R[e_1, e_2, e_3] \text{ is the set of symmetric polynomials.} \\ \text{Look up the fundamental theorem of symmetric polynomials.}
Definition 56. Poset
A partially ordered set (AKA poset)is a pair (P,≤)where P is a set,and ≤ is an order relation on P.However, not every pair of elements needs to be comparable.\text{A partially ordered set (AKA poset)} \\ \text{is a pair } (P, \leq) \\ \text{where } P \text{ is a set}, \\ \text{and } \leq \text{ is an order relation on } P. \\ \text{However, not every pair of elements needs to be comparable.}
Definition 57. Upper Bound of a Poset
Let (P,≤)be a poset and A⊆P.An upper bound for Ais an element m∈P s.t. p≤m  ∀p∈P.\text{Let } (P, \leq) \text{be a poset and } A \subseteq P. \\ \text{An upper bound for } A \\ \text{is an element } m \in P \text{ s.t. } p \leq m \ \ \forall p \in P.
Definition 58. Chain
A chain in (P,≤) is a subset C∈Ps.t. ∀x,y∈C,  x≤y or y≤x.\text{A chain in } (P, \leq) \text{ is a subset } \mathcal{C} \in P \\ \text{s.t. } \forall x, y \in \mathcal{C}, \ \ x \leq y \text{ or } y \leq x.
Guatemamas

Up the road to Santa Cruz la Laguna, Guatemala

You're almost there!