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 ↺ ) → (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 )n∈ 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 ∈X ( ) = 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) ∧ ( X)) → (1, ...,(n) = N) q,G G q τ G ( ) ( ) ∣ ∣ ( ↺ ( = { })) → ( = ∑∈ )
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.
↺
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 k∈ Z ( = ⟨ ⟩) ↺ ∈ (b) σ β β complete factorization O( ) 1= { t ( ) ∶ ∈ } (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. G X G acts on X
↺
43 #action #group #notation #transitive
47. G t X G acts transitively on X
↺
44 #action #homomorphism #symmetric
h 48. (a) α G SX, g αg h (b) B→ (G →SX ↦β G) X X, β g, x B g x action ( ∶ → ) → ( ∶ × → ( ) = ( )( ) = )
45 #action #index #orbit #order #stabilizer
49. G X x G Gx
( ↺ ) → (∣O( )∣ = [ ∶ ])
46 #action #isomorphism #symmetric
50. (a) G SX G X (b) G S ∀ ≤ X ∶ ↺
47 #action #kernel #normal #subgroup
51. (ker action) G
⊴
48 #action #kernel #permutation #representation
52. kernel of the action kernel of the associated permutation represen- tation =
49 #action #orbit #order
53. G X X i xi X k X xi i one x is chosen from each orbit ( ↺i ) → ( = ⊔ O( )) → ((∣ ∣ = ) → (∣ ∣ = ∑ ∣O( )∣)) 54. G X, G k x G
55. (G ↺ X ∣fin∣i=te ) → (∣O( )∣ ∣ ∣ ∣) 1 X N G F g number of orbits ↺ ( g= ) F XO g x g = ∣ ∣ ∑numb(er)o=f fixed by ( ) =
50 #action #translation
56. G G by (left) translations τa x ax
↺ ∶ ↦
51 #adjoin #definition #identity #monoid #semigroup
, if has an identity 57. (a) 1 ⎧ 1 , otherwise ⎪S S S1 = ⎨ (b) ⎪monoid obtained by adjoining an identity to if necessary ⎩⎪S ∪ { } S = S
52 #adjoin #definition #semigroup #zero
, if has a zero 58. (a) 0 ⎧ 0 , otherwise ⎪S S S0 = ⎨ (b) ⎪semigroup obtained by adjoining a zero to if necessary ⎩⎪S ∪ { } S = S
53 #affine #definition #group
59. (a) fa,b R R, fa,b ax b affine map, a, b R, a 0 (b) Aff(1, ) affine group all affine maps ∶ R→ = + = ∈ ≠ (c) group under composition = =
54 #alternating #group #notation
60. An alternating group
=
55 #alternating #index #prime
61. S A6 A6 S p
∃ ≤ ∶ [ ∶ ] =
56 #alternating #order #subgroup
62. A4 12, S A4 S 6
∣ ∣ = ∃ ∈ ∶ ∣ ∣ =
57 #alternating #order #subgroup #symmetric
63. ! S A4 S4 S 12
∃ ( = ) ≤ ∶ ∣ ∣ =
58 #alternating #order #symmetric
1 64. n 3 !An Sn An 2n! ( ≥ ) → (∃ ≤ ∶ ∣ ∣ = )
59 #alternating #simple
s 65. A5 A5
s 66. A6 = A6
67. n= 5 An simple
∀ ≥ ∶ =
60 #alternating #even #permutation #subgroup #symmetric
ev ev 68. An Pn , Pn even permutations of Sn , An subgroup
⊆ = { } =
61 #arrangement #definition #order #set
69. X n arrangement of X list x1, x2, ..., xn with no repetitions xi X (∣ ∣ = ) → ( = ∀ ∈ )
62 #associative #binary operation #definition #identity #monoid #semigroup #set
70. monoid semigroup (set binary operation associative) identity
= + + +
63 #associative #binary operation #definition #semi- group #set
71. S, semigroup non-empty set (S) associative binary operation ( ) S = ( ○) = = + ○
64 #associative #cartesian product #definition #direct product #semigroup
72. (a) i i I collection of semigroups (b) direct product (cartesian product) semigroup S i=I{Si ∶ ∈ } = (c) i ∈ st i s i t ..., s , ...... , t , ...... , s t , ... compo- ∏ S = i i = i i nentwise multiplication associative ( )( ) = ( ) ( ) = ( )( ) = ( ) = =
65 #automorphism #bijection #center #definition #group #homomorphism #kernel #normal #subgroup
73. f a f h G G
↣→ a 74. Au≡t (G ∶ fi→ i)
75. (a)((A)u=t({G), ∶ ∀) } group (b) γ G Aut G , g γ , γ f h ○ = g (c) ker γ Z G ∶ → ( ) ↦ = (d) im γ Aut G = ( ) ⊴ ( )
66 #automorphism #characteristic #definition #subgroup
76. H is characteristic in G (H char G) if
(a) H G (b) σ Aut G σ H H ≤ ∀ ∈ ( ) ∶ ( ) =
67 #automorphism #cyclic #Euler function #group #order
77. G n Aut G φ n
(∣⟨ ⟩∣ = ) → (∣ (⟨ ⟩)∣ = ( ))
68 #automorphism #group #order
78. Aut G 1 G 2
( ( ) = { }) ↔ (∣ ∣ ≤ )
69 #automorphism #group #permutation
79. automorphisms permutations of the set G
=
70 #automorphism #homomorphism #isomorphism #notation
h h h 80. , , f , ϕ , ..., fh, ϕh, ... homomorphism
81. ≃, →f i, ϕi, ... isomorphism
82. f a, ϕa, ... automorphism
71 #automorphism #subgroup #permutation #symmetric
83. Aut G SG
( ) ≤
72 #binary operation #cartesian product #definition #group #set
84. G G G binary operation
85. ∗a ∶ b ×ab → juxtaposition
∗ =
73 #binary operation #definition #group #set #subgroup
86. x, y S x y S S closed under
87. (S∀ubgrou∈p S∶ ∗S,∈ ) → ( = ∗)
(a) 1 S = ( ∗) (b) S is closed under , x, y S xy S ∈ (c) x S x 1 S ∗ ( ∈ ) → ( ∈ ) − ( ∈ ) → ( ∈ )
74 #cancellative #group #semigroup
88. finite cancellative semigroup group
(S = ) → (S = )
75 #cancellative #left cancellative semigroup #left zero semigroup #right #semigroup #zero
89. x, y xy y x, y, z zx zy x y
(∀ ∈ S ∶ = ) → (∀ ∈ S ∶ ( = ) → ( = ))
76 #cardinality #definition #full #map #partial #relation
90. x X xρ y Y xρy , ρ relation
∀(a)∈ x∶ X= {xρ∈ 1∶ }ρ par=tial map from X to Y (b) x X xρ 1 ρ (full) map from X to Y (∀ ∈ ∶ ∣ ∣ ≤ ) → ( = ) (∀ ∈ ∶ ∣ ∣ = ) → ( = )
77 #cartesian product #definition #map #set
91. (a) i I Xi cartesian product set of maps σ I i I Xi such i I iσ X that i. ∈ ∏ ∈ = = ∶ → ⋃ (b) I 1, ..., n finite X finitary ∀ ∈ ∶ ∈ i I i ( = { } = ) → (∏ ∈ = )
78 #center #centralizer #class equation #conjugacy class #definition #group #index #order
92. class equation G Z G G CG xi , i one x is selected from each conjugacy class having more than one ∣ ∣ = ∣i ( )∣ + ∑[ ∶ ( )] element
79 #center #centralizer #subgroup
93. (a) Z G CG G (b) Z G G ( ) = ( ) ( ) ≤
80 #center #commute #conjugation #definition #group
94. (a) Z G g G gx xg, x G (b) zg gz z gzg 1 ( ) = { ∈ ∣ = ∀ ∈ } (c) Z G center of G − set of elements commuting with all the ( = ) ↔ ( = ) elements of G ( ) = = 95. Z G 1 centerless
( ) = { }
81 #center #conjugacy class
96. xG x x Z G
( = { }) ↔ ( ∈ ( ))
82 #center #cyclic #group #quaternion
97. I Z Q
⟨− ⟩ = ( )
83 #center #group #linear
d 0 98. Z GL 2, , d 0 R 0 d ( ( )) = [ ] ≠
84 #center #group #symmetric
99. Z Sn 1 , n 3
( ) = { } ∀ ≥
85 #center #normal #subgroup
100. Z G G
( ) ⊴
86 #center #order
101. Gp 1 Z G 1
(∣ ∣ > ) → ( ( ) ≠ { })
87 #center #order #quaternion #subgroup
102. ! S Z Q Q S 2
∃ ( = ( )) ≤ ∶ ∣ ∣ =
88 #centralizer #commute #conjugation #definition #group
103. G group, A G
1 (a=) CG A ∅g≠ G⊆ gag a, a A (b) gag 1 g ga a−g ( ) = { ∈ ∣ = ∀ ∈ } (c) C A− centralizer of A in G set of elements of G which ( G = ) ↔ ( = ) commute with every element of A ( ) = =
89 #centralizer #conjugacy class #index #order
104. (a) CS s S CG s (b) G S 2 sS sG sS 1 sG ( ) = ∩ ( ) 2 sS conjugacy class of s in S ([ ∶ ] = ) → ((∣ ∣ = ∣ ∣) ∨ (∣ ∣ = ∣ ∣)) G G 105. x G = x G CG x x G
( ∈ ) → (∣ ∣ = [ ∶ ( )]) → (∣ ∣ ∣ ∣ ∣)
90 #centralizer #normalizer #subgroup
106. (a) CG A NG A G
( ) ≤ ( ) ≤
91 #characteristic #cyclic #normal #order #prime #subgroup #Sylow #unique
107. P Sylp G
(a≤) np (1 ) (number of Sylow p-subgroups of G) (b) P G = (c) P char G ⊴ (d) X G, x X x yp X p-group
( ≤ ∀ ∈ ∶ ∣ ∣ = ) → (⟨ ⟩ = )
92 #characteristic #cyclic #subgroup
108. H G H char G
∀ ≤ ⟨ ⟩ ∶
93 #characteristic #normal #order #subgroup #unique
109. (a) H char G G (b) !H G, G n H char G (∀ ) ⊴ (c) K char H H G K G (∃ ≤ ∣ ∣ = ) → ( ) (( ) ∧ ( ⊴ )) → ( ⊴ )
94 #circle #definition #group
110. Circle group (circle S1 of radius 1, center in the origin): S1 z C z 1 , the operation is multiplication.
= { ∈ ∶ ∣ ∣ = }
95 #closed #definition
111. SS S S closed
( ⊆ ) → ( = )
96 #closed #definition #left #ideal #right #semigroup #subset
112. T S
∅(a≠) S⊆T T T left ideal (b) T S T T right ideal ( ⊆ ) ≡ ( = ) (c) ST T S T T ideal two-sided ideal) ( ⊆ ) ≡ ( = ) ( ∪ ⊆ ) ≡ ( = =
97 #closed #definition #left #product #right
113. T S
∅(a≠) S⊆T T T closed under left multiplication (b) T S T T closed under right multiplication ( ⊆ ) → ( = ) (c) ST T S T ( ⊆ ) → ( = ) T closed under both left and right multiplication ( ∪ ⊆ ) → ( = )
98 #closed #definition #semigroup #subsemigroup
114. (a) T T T T closed T subsemigroup (b) T T x, y T xy T (∅ ≠ ≤ S ∶ ⊆ ( )) ≡ ( = ) ( ) ≡ (∀ ∈ ∶ ∈ )
99 #color
115. set of q colors
116. Cn= all n-tuples of colors
X 117. CN q=,G number of q, G -colorings of X ( ) = ( )
100 #color #cycle index #order #symmetric
X 118. X n, G Sn N q,G PG q, ..., q (∣ ∣ = ≤ ) → ( ( ) = ( ))
101 #color #order #symmetric
X 119. (a) N q,G number of q, G -colorings of X G S , X n, q, i 1 σ ci ... ci (b) ( ) =X ( ) i 1 q X N q,G having fr elements of color cr r ( ≤ ∣ ∣ = ∣fC1 ∣ f=2 fq∀ ≥ ∶ = + + ) → the coefficient of c c cq → ( ( ) 1 2 ∀ = = ⋯ )
102 #commutative #opposite #semigroup
120. commutative opp
(S = ) → (S = S)
103 #conjugacy class #group #notation #orbit
121. x xG conjugacy class of x
O( ) = =
104 #conjugacy class #normal #subgroup
G 122. (a) S S i gi G (b) g ⊴ 1⊴ ∀1 ∶ = ⋃ = { }
105 #conjugation #definition #group
c 1 123. αg x gxg conjugation − 124. ga(g 1) ≡ conjugate of a G − 1 125. γg G G, a G γg a∈ gag ; conjugation by g − 126. a,∶b →G, g ∀G ∈ b ∶gag( 1) = a and b are conjugate in G − ( ∈ ∃ ∈ ∶ = ) → ( )
106 #conjugation #definition #group #normal #subgroup
127. ghg 1 H H G normal subgroup − ( ∈ ) ≡ ( ⊴ )
107 #conjugation #definition #group #normalizer
128. (a) gAg 1 gag 1 a A (b) N −A g −G gAg 1 A normalizer of A in G G = { ∣ ∈ } − ( ) = { ∈ ∣ = } =
108 #conjugation #divisibility #index #mod #normalizer #order #prime #subgroup #Sylow
129. G group, p prime, G pαm, p m [4]
(a=) Sylp G = ∣ ∣ = ∣ (b) P Syl G Q p-subgroup of G ( ) ≠p∅ g G Q gP g 1 ; in particular, any two Sylow p- s(u(bg≤roups o(f G))a∧re( co=njug−ate in G )) → → (∃ ∈ ∶ ≤ ) (c) np G np 1 mod p , NG P normalizer, np G NG P , np m ( ) ≡ ≡ ( ) ( ) = = ∣ ∶ ( )∣ ∣
109 #conjugation #group #homomorphism #kernel
130. f h G H
131. x ∶ ke→r f h gxg 1 ker f h − ( ∈ ) → ( ∈ )
110 #conjugation #group #isomorphism #order
132. (a) γg (b) gag 1 gbg 1 , a, b G − − ∣ ∣ = ∣ ∣ ∈
111 #conjugation #normal #subgroup
133. S G g gsg 1 S − ( ⊴ ) ↔ (∀ ∶ ∈ )
112 #connective #notation
134. exclusive or
∨
113 #coprime #cyclic #group #order #power
135. G g , G n gk generator gcd k, n 1
( = ⟨ ⟩ ∣ ∣ = ) → (( = ) ↔ ( ( ) = ))
114 #coprime #integer modulo m #isomorphism
136. m, n 1 Imn Im In
(( ) = ) → ( × )
115 #coset #cyclic #group #quaternion
137. (a) Q I, A, A2, A3, B, BA, BA2, BA3 (b) B A B, BA, BA2, BA3 coset = { } ⟨ ⟩ = { } =
116 #coset #definition #group #index #order
G 138. G S S index of S in G number of cosets ∣ ∣ ∣ ∶ ∣ = ∣ ∣ = =
117 #coset #definition #group #subgroup
139. gS gs (left) coset, gS G
140. Sg = {sg} = (right) coset, Sg⊆ G
= { } = ⊆
118 #coset #equivalence #subgroup
141. s gS
[ ] =
119 #coset #group #notation
142. G H family of cosets of H G