TheoremsDefinitionsArticles

This is a catalog of results I have learned in my Abstract Algebra course, alongside proofs for each.
Apologies in advance as all statements are presented as theorems for my sake.
My favorites got a tombstone for how well they've been put to rest.

Theorem 1. Cancellation Law
Let G be a group. If ab=ac then b=c\text{Let } G \text{ be a group. If } ab = ac \text{ then } b = c

Theorem 2. Uniqueness of Inverse
For any group, the inverse of any element a is unique\text{For any group, the inverse of any element } a \text{ is unique}

Theorem 3. Subgroup Test
Let G be a group and let H⊆G.Then H is a subgroup of G iff H is non-empty and∀a,b∈H we have a−1b∈H\text{Let } G \text{ be a group and let } H \subseteq G. \\ \text{Then } H \text{ is a subgroup of } G \text{ iff } H \text{ is non-empty} \text{ and} \\ \forall a, b \in H \text{ we have } a^{-1}b \in H

Theorem 4. Subgroup Equivalence Relation
Let G be a group and let H≤G.The relation ∀x,y∈Gx−1y∼(x)−1  ⟺  x−1y∈H,is an equivalence relation.\text{Let } G \text{ be a group and let } H \leq G. \\ \text{The relation } \forall x, y \in G \\ x^{-1}y \sim (x)^{-1} \iff x^{-1}y \in H, \\ \text{is an equivalence relation.}

Theorem 5. Left Cosets of x are Equivalence Classes of x
Let H≤G and x−1y∼(x)−1  ⟺  x−1y∈H.Then ∀x∈G, we have [x]={xh ∣ h∈H}.\text{Let } H \leq G \text{ and } x^{-1}y \sim (x)^{-1} \iff x^{-1}y \in H. \\ \text{Then } \forall x \in G, \text{ we have } [x] = \{xh \ | \ h \in H\}.

Theorem 6. Intersection of Subgroups
Let G be a group and let H and K be two subgroups.Then H∩K is also a subgroup of G\text{Let } G \text{ be a group and let } H \text{ and } K \text{ be two subgroups.} \\ \text{Then } H \cap K \text{ is also a subgroup of } G

Theorem 7. Homomorphism Properties
Let ϕ:G→K be a homomorphism.Then ϕ(eG)=eK and ∀a∈G, ϕ(a−1)=(ϕ(a))−1.\text{Let } \phi: G \to K \text{ be a homomorphism.} \\ \text{Then } \phi(e_G) = e_K \text{ and } \forall a \in G, \ \phi(a^{-1}) = (\phi(a))^{-1}.

Theorem 8. Image of a Subgroup is a Subgroup
Let ϕ:G→K be a homomorphism. and let H≤G.Then ϕ(H)≤K.\text{Let } \phi: G \to K \text{ be a homomorphism. and let } H \leq G. \\ \text{Then } \phi(H) \leq K.

Theorem 9. Kernel of a Homomorphism is a Subgroup of The Domain
Let ϕ:G→K be a homomorphism. Then ker⁡(ϕ)≤G\text{Let } \phi: G \to K \text{ be a homomorphism.} \text{ Then } \ker(\phi) \leq G

Theorem 10. Injective Homomorphism
A homomorphism ϕ:G→K is injectiveiff ker⁡(ϕ)={e}\text{A homomorphism } \phi: G \to K \text{ is injective} \\ \text{iff } \ker(\phi) = \{e\}

Theorem 11. Normal Subgroup Properties
Let H≤G. then the following are equivalent:1) H⊴G2) xH∗yH=xyH ∀x,y∈G is well-defined on the coset space G/H3) ∀x∈G we have xH=Hx.\text{Let } H \leq G. \text{ then the following are equivalent:} \\ \quad \text{1) } H \trianglelefteq G \\ \quad \text{2) } xH * yH = xyH \ \forall x, y \in G \\ \quad \quad \ \text{is well-defined on the coset space G/H} \\ \quad \text{3) } \forall x \in G \text{ we have } xH = Hx.

Theorem 12. Quotient Group
Let H⊴G. Then G/H with coset multiplication is a group\text{Let } H \trianglelefteq G. \text{ Then } G/H \text{ with coset multiplication is a group}

Theorem 13. First Isomorphism Theorem
Let G and K be groups,and let ϕ:G→K be a homomorphism.Then ker⁡(ϕ)⊴G, and G/ker⁡(ϕ)≅Im(ϕ) canonically.\text{Let } G \text{ and } K \text{ be groups,} \\ \text{and let } \phi: G \to K \text{ be a homomorphism.} \\ \text{Then } \ker(\phi) \trianglelefteq G, \text{ and } G/\ker(\phi) \cong Im(\phi) \text{ canonically.}

Theorem 14. The Group of Automorphisms
The set Aut(G) with composition is a group.\text{The set Aut}(G) \text{ with composition is a group.}

Theorem 15. Inner Automorphism
Let g∈G.Define ϕg:G→G such that ϕg(x)=gxg−1  ∀x∈G.ϕg is an automorphism.\text{Let } g \in G. \\ \text{Define } \phi_g: G \to G \text{ such that } \phi_g(x) = gxg^{-1} \ \ \forall x \in G. \\ \phi_g \text{ is an automorphism}.

Theorem 16. Subscript of Inner Automorphisms is Homomorphic
The map Φ:G→Aut(G) defined by Φ(g)=ϕg  ∀g∈G, is a homomorphism.\text{The map } \Phi: G \to \text{Aut}(G) \text{ defined by } \\ \Phi(g) = \phi_g \ \ \forall g \in G, \text{ is a homomorphism.}

Theorem 17. The Inner Automorphisms Are A Normal Subgroup of Automorphisms
Inn(G)⊴Aut(G)\text{Inn}(G) \trianglelefteq \text{Aut}(G)

Theorem 18. Sets of the Same Cardinality have Isomorphic Permutation Groups
Let X and Y be sets.If ∣X∣=∣Y∣, then SX≅SY.\text{Let X and Y be sets.} \\ \text{If } |X| = |Y|, \text{ then } S_X \cong S_Y.

Theorem 19. Left Multiplication is a Bijective Permutation that is the Image of any Group
Let G be a group.For each a∈G define fa:G→G by fa(x)=ax.Then fa is bijective  ∀a∈G.Define j:G→SG by j(a)=fa.Then j is an injective homomorphism.\text{Let } G \text{ be a group.} \\ \text{For each } a \in G \text{ define } f_a: G \to G \text{ by } f_a(x) = ax. \\ \text{Then } f_a \text{ is bijective } \ \forall a \in G. \\ \text{Define } j: G \to S_G \text{ by } j(a) = f_a. \\ \text{Then } j \text{ is an injective homomorphism.}

Theorem 20. Cayley's Theorem
Every Group is Isomorphic to a Subgroup of a Symmetric Group\text{Every Group is Isomorphic to a Subgroup of a Symmetric Group}

Theorem 21. In Finite Groups, Elements Have Finite Order (PP1 Q10)
Let (G,∗) be a finite group. Prove that for every a∈G∃ a natural number n≥1 such that an=e.\text{Let } (G,*) \text{ be a finite group. Prove that for every } a \in G \\ \exists \text{ a natural number } n \geq 1 \text{ such that } a^n = e.

Theorem 22. Cycles are Equivalence Relations
Let σ∈Sn be fixed.Then the relation on {1,2,3,...,n},∀ 1≤i,j≤n, i∼j  ⟺  σk(i)=j for some k∈Z,is an equivalence relation.\text{Let } \sigma \in S_n \text{ be fixed.} \\ \text{Then the relation on } \{1, 2, 3, ..., n\}, \\ \forall \ 1 \leq i, j \leq n, \ i \sim j \iff \sigma^{k}(i) = j \text{ for some } k \in \Z, \\ \text{is an equivalence relation.}

Theorem 23. Equivalence classes of a permutation times a transposition differs by 1
Let n∈N such that 2≤n, and σ∈Sn.Let 1≤i≠j≤n, and τ=(i j).Then the number of equivalence classes ofthe relations defined by σ and τσ differs by 1.\text{Let } n \in \N \text{ such that } 2 \leq n, \text{ and } \sigma \in S_n. \\ \text{Let } 1 \leq i \not= j \leq n, \text{ and } \tau = (i \ j). \\ \text{Then the number of equivalence classes of} \\ \text{the relations defined by } \sigma \text{ and } \tau \sigma \text{ differs by 1}.

Theorem 24. Any Permutation is the Product of Transpositions
Let n≥2. Any permutation σ∈Snis a product of transpositions.\text{Let } n \geq 2. \text{ Any permutation } \sigma \in S_n \\ \text{is a product of transpositions.}

Theorem 25. A Permutation is either Odd or Even
A permutation σ∈Sn can’t be writtenas a product of an even number of transpositionsand an odd number of transpositions.\text{A permutation } \sigma \in S_n \text{ can't be written} \\ \text{as a product of an even number of transpositions} \\ \text{and an odd number of transpositions.}

Theorem 26. Parity Homomorphism
Let n≥2. Defineϕ:Sn→Z2 by ϕ(σ)={0,if σ is even1,if σ is oddThis is a homomorphism.\text{Let } n \geq 2. \text{ Define} \\ \phi: S_n \to \Z_2 \text{ by } \\ \phi(\sigma) = \begin{cases} 0, & \text{if } \sigma \text{ is even} \\ 1, & \text{if } \sigma \text{ is odd} \\ \end{cases} \\ \text{This is a homomorphism.}

Theorem 27. The Alternating Group is a Normal Subgroup of the Symmetric Group
An⊴SnA_n \trianglelefteq S_n

Theorem 28. Cosets of H have the same Cardinality as H
Let G be a finite group,and let H≤G.For every x,y∈G, the map f:xH→yHdefined by f(xh)=yh  ∀h∈His a bijection.\text{Let G be a finite group,} \text{and let } H \leq G. \\ \text{For every } x, y \in G, \text{ the map } f: xH \to yH \\ \text{defined by } f(xh) = yh \ \ \forall h \in H \\ \text{is a bijection.}

Theorem 29. Pretty Much Lagrange's Theorem
Let G be a finite group, and H≤G.Then o(G)=o(H)[G:H].([G:H]=∣G:H∣= Number of distinct left cosets of H in G)\text{Let } G \text{ be a finite group, and } H \leq G. \\ \text{Then } o(G) = o(H) [G:H]. \\ ([G:H] = |G:H| = \text{ Number of distinct left cosets of H in G})

Theorem 30. Actually Lagrange's Theorem
Let G be a finite group, and H≤G.Then o(H) divides o(G).\text{Let } G \text{ be a finite group, and } H \leq G. \\ \text{Then } o(H) \text{ divides } o(G).

Theorem 31. Order of the Alternating Group
∣An∣=n!2  for n≥2.|A_n| = \frac{n!}{2} \ \ \text{for } n \geq 2.

Theorem 32. Direct Product of Groups is a Group
The set G=G1×G2 with operation  defined by (g1,g2)⋅(h1,h2)=(g1h1,g2h2) for all g1,h1∈G1,g2,h2∈G2 is a group.\text{The set } G = G_1 \times G_2 \text{ with operation } \\ \text{ defined by } (g_1, g_2) \cdot (h_1, h_2) = (g_1h_1, g_2h_2) \\ \text{ for all } g_1, h_1 \in G_1, g_2, h_2 \in G_2 \text{ is a group.}

Theorem 33. Direct Product Homomorphisms
Let G1 and G2 be groups,and let G=G1×G2.For each i=1,2define ρ1:G1→G by ρ1(g1)=(g1,e)  ∀g1∈G1,and π1:G→G1 by π1((g1,g2))=g1.Then ρ1 is an injective homomorphism,π1 is a surjective homomorphism,ker⁡(π1)={e}×G2,and G/ker⁡(π1)≅G1.These same homomorphisms can be defined for G2.Just swap the 1’s and 2’s and flip ρso ρ2(g2)=(e,g2)\text{Let } G_1 \text{ and } G_2 \text{ be groups,} \\ \text{and let } G = G_1 \times G_2. \\ \text{For each } i = 1, 2 \\ \text{define } \rho_1: G_1 \to G \text{ by } \rho_1(g_1) = (g_1, e) \ \ \forall g_1 \in G_1, \\ \text{and } \pi_1: G \to G_1 \text{ by } \pi_1((g_1, g_2))= g_1. \\ \text{Then } \rho_1 \text{ is an injective homomorphism,} \\ \pi_1 \text{ is a surjective homomorphism}, \\ \ker(\pi_1) = \{e\} \times G_2, \\ \text{and } G/\ker(\pi_1) \cong G_1. \\ \text{These same homomorphisms can be defined for } G_2. \\ \text{Just swap the 1's and 2's and flip } \rho \\ \text{so } \rho_2(g_2) = (e, g_2)

Theorem 34. Direct Product Observation
Not Exam 3 Material\text{Not Exam 3 Material}

Theorem 35. Conditions for G isomorphic to G1 X G2
Not Exam 3 Material\text{Not Exam 3 Material}

Theorem 36. k multiple of order iff a^k = e
Not Exam 3 Material\text{Not Exam 3 Material}

Theorem 37. Cardinality and Order are the same in Cyclic Groups
Not Exam 3 Material\text{Not Exam 3 Material}

Theorem 38. Cyclic Groups with Same Order are Isomorphic
Let G and K be cyclic groups.Then we have G≅K  ⟺  ∣G∣=∣K∣. Not Exam 3 Material\text{Let } G \text{ and } K \text{ be cyclic groups.} \\ \text{Then we have } G \cong K \iff |G| = |K|. \\ \ \\ \text{Not Exam 3 Material}

Theorem 39. All Cyclic Groups are isomorphic to Z
Not Exam 3 Material\text{Not Exam 3 Material}

Theorem 40. Zm X Zn isomorphic to Zmn iff gcd(m,n) = 1
Not Exam 3 Material\text{Not Exam 3 Material}

Theorem 41. Zn is a ring
Let n∈N.Then (Zn,+,⋅) is a ring where [k]⋅[ℓ]=[kℓ].\text{Let } n \in \N. \\ \text{Then } (\Z_n, +, \cdot) \text{ is a ring where } [k] \cdot [\ell] = [k \ell]. \\

Theorem 42. Algebraic Properties of Rings
Let R be a ring.1) r⋅0=0⋅r=0  ∀r∈R.2) Let R be unital. Then r⋅(−1)=−r  ∀r∈R.\text{Let } R \text{ be a ring.} \\ \quad 1) \ r \cdot 0 = 0 \cdot r = 0 \ \ \forall r \in R. \\ \quad 2) \text{ Let } R \text{ be unital. Then } r \cdot (-1) = -r \ \ \forall r \in R.

Theorem 43. Cancellation Law for Rings
If R is an integral domain anda,b,c∈R s.t. ab=ac,then either a=0 or b=c.\text{If R is an integral domain and} \\ a,b,c \in R \text{ s.t. } ab = ac, \\ \text{then either } a = 0 \text{ or } b = c.

Theorem 44. Image of Ring Homomorphism Is Subring of Codomain
Let ϕ:R→S be a ring homomorphism.Then ϕ(R)≤S.\text{Let } \phi: R \to S \text{ be a ring homomorphism.} \\ \text{Then } \phi(R) \leq S.

Theorem 45. Kernel of Ring Homomorphism is Subring of Domain
Let ϕ:R→S be a ring homomorphism.Then ker⁡(ϕ)≤R.\text{Let } \phi: R \to S \text{ be a ring homomorphism}. \\ \text{Then } \ker(\phi) \leq R.

Theorem 46. R/I is a Ring only when I is an Ideal
The formula (x+I)⋅(y+I)=xy+Iis a well-defined binary operation on R/Iiff I is an ideal in R.Moreover, in this case,(R/I,+,⋅) is a ring,called the quotient ring of R and I.\text{The formula } (x+I) \cdot (y+I) = xy+I \\ \text{is a well-defined binary operation on } R/I \\ \text{iff } I \text{ is an ideal in } R. \\ \text{Moreover, in this case,} (R/I, +, \cdot) \text{ is a ring,} \\ \text{called the quotient ring of } R \text{ and } I.

Theorem 47. FIT for Rings
Let ϕ:R→S be a ring homomorphism.Then ker⁡(ϕ)⊴R,and R/ker⁡(ϕ)≅ϕ(R) canonically.\text{Let } \phi: R \to S \text{ be a ring homomorphism.} \\ \text{Then } \ker(\phi) \trianglelefteq R, \\ \text{and } R/\ker(\phi) \cong \phi(R) \text{ canonically.}

Theorem 48. Proper Ideals can't contain the Multiplicative Identity
Let R be unital and I⊴R.I is proper iff 1∉I.\text{Let } R \text{ be unital and } I \trianglelefteq R. \\ \text{I is proper iff } 1 \not \in I.

Theorem 49. Zorn's Lemma
Suppose (P,≤) is a non-empty posetwith the property that every chain in Phas an upper bound.Then P contains a maximal element.\text{Suppose } (P, \leq) \text{ is a non-empty poset} \\ \text{with the property that every chain in } P \\ \text{has an upper bound}. \\ \text{Then } P \text{ contains a maximal element.}

Theorem 50. Commutative Unital Rings have Maximal Ideals
Every commutative unital ring R has a maximal ideal.\text{Every commutative unital ring } R \text{ has a maximal ideal.}

Theorem 51. Properties of Ideals in Unital Rings
Let R be a unital ring and I⊴R.1) I=R2) 1∈I1) I contains an invertible element.\text{Let } R \text{ be a unital ring and } I \trianglelefteq R. \\ \quad \text{1) } I = R \\ \quad \text{2) } 1 \in I \\ \quad \text{1) } I \text{ contains an invertible element.}

Theorem 52. Simple Unital Commutative Rings are Fields
Let R be a unital commutative ring.Then R is a field.iff the only ideals of Rare {0} and R.\text{Let } R \text{ be a unital commutative ring.} \\ \text{Then } R \text{ is a field.} \\ \text{iff the only ideals of } R \\ \text{are } \{0\} \text{ and } R.

Theorem 53. Ideals in Chain are the Same iff Quotient Rings are the Same
Let I⊴J⊴P⊴R.Then J=P iff J/I=P/J.\text{Let } I \trianglelefteq J \trianglelefteq P \trianglelefteq R. \\ \text{Then } J = P \text{ iff } J/I = P/J.

Theorem 54. Surjective Ring Homomorphism Preserves Ideals
Let ϕ:R→S be a surjective ring homomorphism. if I⊴R, then ϕ(I)⊴S.\text{Let } \phi: R \to S \text{ be a surjective ring homomorphism.} \\ \text{ if } I \trianglelefteq R, \text{ then } \phi(I) \trianglelefteq S.

Theorem 55. Isaiah Original (From Office Hours)
Let ϕ:R→S be a ring homomorphism.If J⊴S, then ϕ−1(J)⊴R. Preimage: ϕ−1(J)={r∈R ∣ ϕ(r)∈J}.\text{Let } \phi: R \to S \text{ be a ring homomorphism.} \\ \text{If } J \trianglelefteq S, \text{ then } \phi^{-1}(J) \trianglelefteq R. \\ \ \\ \text{Preimage: } \phi^{-1}(J) = \{ r \in R \ | \ \phi(r) \in J \}.

Theorem 56. Natural Homomorphism to Quotient Ring
Let I⊴R.Then the map π:R→R/Idefined by π(r)=r+I  ∀r∈Ris a surjective ring homomorphism.\text{Let } I \trianglelefteq R. \\ \text{Then the map } \pi: R \to R/I \\ \text{defined by } \pi(r) = r + I \ \ \forall r \in R \\ \text{is a surjective ring homomorphism.}

Theorem 57. Ideals of Quotient Rings Can be Quotient Rings of Ideals
Let R be a ring and I⊴R.If I⊴J⊴R, then J/I⊴R/I.Conversely, if K⊴R/I,then ∃I⊴J⊴R s.t. K=J/I. This proof is different from the one in the notes.I prefered this method from Theorem 55.\text{Let R be a ring and } I \trianglelefteq R. \\ \text{If } I \trianglelefteq J \trianglelefteq R, \text{ then } J/I \trianglelefteq R/I. \\ \text{Conversely, if } K \trianglelefteq R/I, \\ \text{then } \exists I \trianglelefteq J \trianglelefteq R \text{ s.t. } K = J/I. \\ \ \\ \text{This proof is different from the one in the notes.} \\ \text{I prefered this method from Theorem 55.}

Theorem 100. Maximal Ideal iff Quotient Ring is a Field

Theorem 1000. Generated Sets are Equivalent when their Generators are divisible of each other.
Let a,b∈R.Then ⟨a⟩=⟨b⟩ iff b=au,for some invertible u∈R.\text{Let } a, b \in R. \\ \text{Then } \langle a \rangle = \langle b \rangle \text{ iff } b = au, \\ \text{for some invertible } u \in R.

Theorem 1001. Generators are only Irreducible when the Generating Set is Prime
a∈R is irreducible iff ⟨a⟩ is prime.a \in R \text{ is irreducible iff } \langle a \rangle \text{ is prime.}

Theorem 1002. An ideal of Z is only maximal when its generator is prime
Let n∈N.Then n is prime iff nZ⊴Z is maximal.\text{Let } n \in \N. \\ \text{Then } n \text{ is prime iff } n \Z \trianglelefteq \Z \text{ is maximal}.

Theorem 1003. Generalization of Previous Theorem
In any Principal Integral Domain (PID),every non-zero prime ideal is maximal.\text{In any Principal Integral Domain (PID),} \\ \text{every non-zero prime ideal is maximal.}

Theorem 1004. Ott's 2nd Theorem
Let R be an ID and I⊴R.Then I is prime iff R/I is an ID.\text{Let } R \text{ be an ID and } I \trianglelefteq R. \\ \text{Then } I \text{ is prime iff } R/I \text{ is an ID.}

Theorem 1005. Sort of Fundamental Theorem of Algebra
Let p(x)∈R[x] be a non-constant polynomial.Then ∃ a field F containing an isomorphic copy of Rand α∈F such that p(α)=0.\text{Let } p(x) \in \R[x] \text{ be a non-constant polynomial.} \\ \text{Then } \exists \text{ a field } \mathbb{F} \text{ containing an isomorphic copy of } \R \\ \text{and } \alpha \in \mathbb{F} \text{ such that } p(\alpha) = 0.

Fun times in algebra