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Open Mathematics Knowledge Base Open Journal of Mathematics and Physics | Volume 2, Article 103, 2020 | ISSN: 2674-5747 https://doi.org/10.31219/osf.io/evq8a | published: 13 May 2020 | https://ojmp.org DZ [knowledge base] Diamond Open Access Open Mathematics Knowledge Base Open Mathematics Collaboration∗† August 4, 2020 Abstract We present a list of mathematical results for future implementation in a digital Open Mathematics Knowledge Base. keywords: abstract algebra, pure mathematics, knowledge base The most updated version of this paper is available at https://osf.io/evq8a/download Introduction A. [1–7] B. OMKB Open Mathematics Knowledge Base C. This art=icle is constantly being updated. D. OMKB 444 mathematical entries (309 pages) E. 1 entry = notation or definition or proposition or theorem = ∗All authors with their affiliations appear at the end of this paper. †Corresponding author: [email protected] | Join the Open Mathematics Collaboration 1 Overview F. [8,9] 2 How to build the OMKB G. Write the result of a mathematical proposition/theorem in the fewest number of symbols as possible, without losing information. H. Use a very intuitive and compact notation. I. Assign all important tags associated with each result. J. There are mainly two categories of tags: (a) notations/definitions, (b) results. K. Important: some tags are contained in more general tags. L. For example, the definition of center contains the tag abelian. In this case, while within the notation and/or the definition tags, one should include both, abelian and center. M. On the other hand, considering the results for center, there is no need to add the tag abelian since it is implicitly understood. N. In summary, for (J.a), we consider all tags (general and specific); while in (J.b), we consider only the most specific tags. 3 Important: guidelines for advanced search O. In order to maximize the efficiency while using this document, note the following guidelines for advanced search in Acrobat Reader. P. Acrobat Reader Preferences Search Range of words for proximity searches 10 → → → Q. Acrobat Reader o≈pen the search window advanced settings show more options select a folder where the PDF is located check the proxim→ity box choose match all→of the words → → → → 4 #abelian #boolean #definition #group #identity #inverse #symmetric difference 1. A, B X, A B A B B A symmetric difference; X Boolean (abelian) group family of all subsets of X (operation sym⊆metric +diffe=re(nce−) ) ∪ ( − ) = B( ) = = =(a) A B B A (b) identity + = + (c) A 1 A, A A ∅ = − = + = ∅ 5 #abelian #center #group 2. gg g g Z G G " " ( = ) ↔ ( ( ) = ) 6 #abelian #definition #group 3. gg g g G GAb " " ( = ) → ( = ) 7 #abelian #definition #group #parity 4. P, , P even, odd abelian parity group P(a=) {even+} eve=n{ even o}d=d odd (b) even odd odd odd even + = = + + = = + 8 #abelian #group 5. Additive abelian groups: Z, Q, R, C. 1 6. Multiplicative abelian groups: Q , R , C , S , Γn. # # # 7. The plane R R is an additive abelian group (vector addition). × 9 #abelian #group #homomorphism #inverse 8. f h G H 1 h 9. g1∶g2 →g2g1 f G G, f g g f g ! ( = ) → ( ∶ → ( ) = = ( )) 10 #abelian #group #notation 10. GAb Abelian group 11 #abelian #group #order 11. G G 6 ab ba 12. (g∀g ∶ ∣ g ∣g< ) → ( = ) " " ∣ ∣ = ∣ ∣ 12 #abelian #group #power 13. g g2 1 abelian (∀ ∶ = ) → ( ) 13 #abelian #normal #subgroup 14. gg g g S G S G " " ( = ) → (( ≤ ) → ( ⊴ )) 14 #abelian #order 15. gg g g " " ∣ ∣ = ∣ ∣ 15 #abelian #order #prime 16. G p2 gg g g " " (∣ ∣ = ) → ( = ) 16 #abelian #order #prime #simple 17. GAb Gs G p ( = ) ↔ (∣ ∣ = ) 17 #action #afford #definition #group #homomorphism #induce #permutation #representation 18. G group, A h (a=) permutation≠ ∅representation of G G SA (b) action of G on A affords/induces the associated permutation rep- ∶ → resentation of G 18 #action #associative #cartesian product #definition #group #identity 19. G X if (a↺) G group, X set (b) α G X X = = (c) α α α , α x gx ∃g ∶ g× g→g g (d) α 1 ! iden! tity function 1 ○ X = ( ) ≡ (e) g g x gg x = = " " ( ) = ( ) 19 #action #associativity #definition #group #identity #set 20. G G, group, Ω , points elements of Ω α g rule that determines a new element of Ω α =g( de○fi)n=es an action o≠f∅G on Ω (G=acts on Ω) if ⋅ = (⋅i) =α 1 α, α Ω; (ii) α g h α gh , α Ω, g, h G. ⋅ = ∀ ∈ ( ⋅ ) ⋅ = ⋅ ( ) ∀ ∈ ∀ ∈ 20 #action #bijection #group #homomorphism h 21. G, A α actions G A homomorphisms f G SA ∀ ≠ ∅ ∃ ∶ ( ↺ ) ↣→ ( ∶ → ) 21 #action #centralizer #conjugacy class #conjugation #definition #group #orbit #stabilizer c 1 22. (a) G G, αg x g aga a G (b) αc x gxg 1 conjugation ! ( g ↺ ( )) → (O( ) = { ∶ ∈ }) (c) gG g !conjugacy class of g ( ) ≡ (d) G stabilizer of x x ≡ O( ) = (e) C x G g G gxg 1 x G G = x (f) C g G, C g centr!alizer of g in G G( ) ≡ = {G ∈ ∶ = ∈ } ( ) ≤ ( ) = 22 #action #conjugation 1 23. G G by conjugation, g, αg G G, αg x gxg . ! ↺ ∀ ∶ → ( ) = 23 #action #conjugation #coset #orbit #stabilizer #translation gH 24. (a) G G H, τa gH GH (b) G H gHg 1 ( g ↺ 6 ) ! (O( ) = ) − = 24 #action #color #orbit 25. (a) G X 1, ..., n G , g c1, ..., cn (b) g g c , ..., c c , ..., c ( / (1 = {n }g1)) ! (gn / C * ( )) (c) c , ..., c n * 1 ) ( n ) = ( ) (d) c , ..., c q, G -coloring of X ( 1 )n4 C O( ) = ( ) 25 #action #color #order #symmetric n n 26. τ Sn, F τ number of elements in fixed by τ t τ numbCer of cycles in the complete factorization of τ 4X ( ) = C N q,G number of q, G -colorings of X ( ) = (a( ) )F= n τ t τ ( )F n τ qt τ C C ( ) X 1 t τ (b) (finit(e G) 7 ( X)) ! (1, ...,(n) = N) q,G G q τ G ( ) ( ) ∣ ∣ ( / ( = { })) ! 8 = 9∈ : 26 #action #conjugation #definition #group #orbit 27. a and b are conjugate in G a are in the same orbit of G acting on itself by conjugation ( ) ' ( 28. orbits of G acting on itself by c)onjugation conjugacy classes of G = 27 #action #conjugation #orbit #order #stabilizer 1 29. (a) G X Ggx gGxg (b) G, X finite − ( ↺ ) → ( = ) G X x, y lie same orbit Gy Gx = (( ↺ ) ∧ ( )) → (∣ ∣ = ∣ ∣) 28 #action #coset #definition #group #orbit #stabilizer 30. x orbit of G containing x X, Gx stabilizer of x G X x gx X Gx g gx x G O( ) = ∈ = ( ↺ ) → (O( ) = { } ⊆ ) → ( = { ∶ = } ≤ ) 29 #action #coset #translation 31. G G H by (left) translation. ↺ 6 30 #action #cyclic #factorization #orbit #order #power #stabilizer #symmetric 32. X 1, 2, ..., n , σ Sn, G σ X, i X (a=) { i σ}k i k4 Z ( = ; <) / 4 (b) σ β β complete factorization O( ) 1= { t ( ) ) 4 } (c) i i is moved by σ = 0 = = (d) β i i ...i j= 0 1 r 1 k (e) k r 1 i − σ i = ( k ) 0 (f) i i , i , ..., i * < $ 0 ) 1 = r(1 ) (g) i r − O( ) = { } (h) l symbol +O( )+ = σ fixes l Gl G = σ moves l Gl G ( ) ! ( = ) ( ) ! ( < ) 31 #action #definition 33. (a) α X Y Z αx Y Z (b) α y α x, y ( x ∶ × → ) ≡ ( ∶ → ) (c) α y one-parameter family of functions (xi ) = ( ) 34. α G{ X( )}X= action of G on X ∶ × → 32 #action #definition #equivalence #group #orbit #transitive 35. G acts on A (a) a ≠g∅a g G equivalence class orbit of G containing a O( ) = { ⋅ ∶ ∈ } = = (b) ! a a, b A, g G, a g b action of G on A transitive (∃ O( ) ≡ ( ∈ ∃ ∈ = ⋅ )) → ( = ) 33 #action #definition #faithful #group #identity #kernel #trivial 36. action faithful kernel identity ( = ) ≡ ( = ) 34 #action #definition #fix #stabilizer 37. (a) g a action (b) for each a A, G g G g a a stabilizer of a in G set ⋅ = a of all elements of G that fix the element a ∈ = { ∈ ∶ ⋅ = } = = 35 #action #definition #group #left 38. (a) G acts on G by left multiplication (b) a, g G (c) g g a ga ∈ ∀ ∶ ⋅ = 36 #action #definition #group #orbit #transitive 39. x X x X G t X 40. ((∃a) ∈ ! ∶ Ox( ) = G) → t(X ↺ ) (b) ! x x, y X g G x gy (∃ O( )) → ( ↺ ) (∃ O( )) ≡ (∀ ∈ ∶ ∃ ∈ ∶ = ) 37 #action #definition #group #permutation #transitive 41. (a) G permutation group, Ω set (b) G Ω = = (c) g x , x , ..., x y , y , ..., y G k-transitive ↺ 1 2 k 1 2 k (d) x g y , x g y , ..., x g y (∃1 ∶ ( 1 2 2) ↦ ( k k )) → ( = ) (e) x x , y y i = j i =j = (f) G k-transitive G k 1 -transitive ≠ ≠ (g) G n-transitive, n 1 G multiply transitive ( = ) → ( = ( − ) ) (h) 1-transitive transitive ( = > ) → ( = ) (i) 2-transitive doubly transitive = (j) 3-transitive triply transitive = = 38 #action #definition #homomorphism #kernel #permutation #representation #symmetric #trivial 42. (a) G G, group (b) G acts on Ω = ( ○) = (c) σg Ω Ω, (d) α σ α g action rule that determines a new element of Ω ∶ g → (e) σ Sym Ω ( g ) = ⋅ = = h (f) g ∈ σg ( p)ermutation representation (g) g G α g α, α Ω ker action G ( ↦ ) = { ∈ ∶ ⋅ = ∀ ∈ } = ⊴ 39 #action #definition #kernel #trivial 43. (a) G acts on A (b) kernel of the action g G g a a, a A set of elements of G that act trivially on every element of A = { ∈ ∶ ⋅ = ∀ ∈ } = 40 #action #equivalence #orbit 44. (a) G X g y gx xReqy (b) x y X yR x gx x ( ↺ ) → ((∃ eq∶ = ) → ( )) [ ] = { ∈ ∶ } = { } = O( ) 41 #action #group #conjugation #definition 45. G acts on G by conjugation (a) a, g G (b) g g a gag 1 ∈ − ∀ ∶ ⋅ = 42 #action #group #notation 46.
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