Home Metamath Proof ExplorerTheorem List (p. 178 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqussub 17701 Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (-g𝐻)       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋𝑉𝑌𝑉) → ([𝑋](𝐺 ~QG 𝑆)𝑁[𝑌](𝐺 ~QG 𝑆)) = [(𝑋 𝑌)](𝐺 ~QG 𝑆))

Theoremlagsubg2 17702 Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝑋 = (Base‘𝐺)    &    = (𝐺 ~QG 𝑌)    &   (𝜑𝑌 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ Fin)       (𝜑 → (#‘𝑋) = ((#‘(𝑋 / )) · (#‘𝑌)))

Theoremlagsubg 17703 Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. This is Metamath 100 proof #71. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝑋 = (Base‘𝐺)       ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∥ (#‘𝑋))

10.2.4  Elementary theory of group homomorphisms

Syntaxcghm 17704 Extend class notation with the generator of group hom-sets.
class GrpHom

Definitiondf-ghm 17705* A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)
GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔[(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥𝑤𝑦𝑤 (𝑔‘(𝑥(+g𝑠)𝑦)) = ((𝑔𝑥)(+g𝑡)(𝑔𝑦)))})

Theoremreldmghm 17706 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom GrpHom

Theoremisghm 17707* Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))

Theoremisghm3 17708* Property of a group homomorphism, similar to ismhm 17384. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))

Theoremghmgrp1 17709 A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)

Theoremghmgrp2 17710 A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)

Theoremghmf 17711 A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)

Theoremghmlin 17712 A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝑋 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹𝑈) (𝐹𝑉)))

Theoremghmid 17713 A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝑌 = (0g𝑆)    &    0 = (0g𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) = 0 )

Theoremghminv 17714 A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝐵 = (Base‘𝑆)    &   𝑀 = (invg𝑆)    &   𝑁 = (invg𝑇)       ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋𝐵) → (𝐹‘(𝑀𝑋)) = (𝑁‘(𝐹𝑋)))

Theoremghmsub 17715 Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝐵 = (Base‘𝑆)    &    = (-g𝑆)    &   𝑁 = (-g𝑇)       ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)𝑁(𝐹𝑉)))

Theoremisghmd 17716* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &   (𝜑𝑆 ∈ Grp)    &   (𝜑𝑇 ∈ Grp)    &   (𝜑𝐹:𝑋𝑌)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))       (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremghmmhm 17717 A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremghmmhmb 17718 Group homomorphisms and monoid homomorphisms coincide. (Thus, GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))

Theoremghmmulg 17719 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    × = (.g𝐻)       ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))

Theoremghmrn 17720 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

Theorem0ghm 17721 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0g𝑁)    &   𝐵 = (Base‘𝑀)       ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁))

Theoremidghm 17722 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))

Theoremresghm 17723 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑈 GrpHom 𝑇))

Theoremresghm2 17724 One direction of resghm2b 17725. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝑋 ∈ (SubGrp‘𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremresghm2b 17725 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))

Theoremghmghmrn 17726 A group homomorphism from 𝐺 to 𝐻 is also a group homomorphism from 𝐺 to its image in 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.)
𝑈 = (𝑇s ran 𝐹)       (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑈))

Theoremghmco 17727 The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))

Theoremghmima 17728 The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹𝑈) ∈ (SubGrp‘𝑇))

Theoremghmpreima 17729 The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (𝐹𝑉) ∈ (SubGrp‘𝑆))

Theoremghmeql 17730 The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹𝐺) ∈ (SubGrp‘𝑆))

Theoremghmnsgima 17731 The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑌 = (Base‘𝑇)       ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹𝑈) ∈ (NrmSGrp‘𝑇))

Theoremghmnsgpreima 17732 The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (𝐹𝑉) ∈ (NrmSGrp‘𝑆))

Theoremghmker 17733 The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
0 = (0g𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝑆))

Theoremghmeqker 17734 Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)
𝐵 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &    = (-g𝑆)       ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) = (𝐹𝑉) ↔ (𝑈 𝑉) ∈ 𝐾))

Theorempwsdiagghm 17735* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ Grp ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌))

Theoremghmf1 17736* Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    0 = (0g𝑆)    &   𝑈 = (0g𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1𝑌 ↔ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )))

Theoremghmf1o 17737 A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)       (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))

Theoremconjghm 17738* Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝐹 = (𝑥𝑋 ↦ ((𝐴 + 𝑥) 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐹 ∈ (𝐺 GrpHom 𝐺) ∧ 𝐹:𝑋1-1-onto𝑋))

Theoremconjsubg 17739* A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺))

Theoremconjsubgen 17740* A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 ≈ ran 𝐹)

Theoremconjnmz 17741* A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))    &   𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑁) → 𝑆 = ran 𝐹)

Theoremconjnmzb 17742* Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))    &   𝑁 = {𝑦𝑋 ∣ ∀𝑧𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}       (𝑆 ∈ (SubGrp‘𝐺) → (𝐴𝑁 ↔ (𝐴𝑋𝑆 = ran 𝐹)))

Theoremconjnsg 17743* A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   𝐹 = (𝑥𝑆 ↦ ((𝐴 + 𝑥) 𝐴))       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋) → 𝑆 = ran 𝐹)

Theoremqusghm 17744* If 𝑌 is a normal subgroup of 𝐺, then the "natural map" from elements to their cosets is a group homomorphism from 𝐺 to 𝐺 / 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))    &   𝐹 = (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))       (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Theoremghmpropd 17745* Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))       (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))

10.2.5  Isomorphisms of groups

Syntaxcgim 17746 The class of group isomorphism sets.
class GrpIso

Syntaxcgic 17747 The class of the group isomorphism relation.
class 𝑔

Definitiondf-gim 17748* An isomorphism of groups is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group operation. (Contributed by Stefan O'Rear, 21-Jan-2015.)
GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})

Definitiondf-gic 17749 Two groups are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic groups share all global group properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑔 = ( GrpIso “ (V ∖ 1𝑜))

Theoremgimfn 17750 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
GrpIso Fn (Grp × Grp)

Theoremisgim 17751 An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))

Theoremgimf1o 17752 An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐵1-1-onto𝐶)

Theoremgimghm 17753 An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Theoremisgim2 17754 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 21610. (Contributed by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))

Theoremsubggim 17755 Behavior of subgroups under isomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐵 = (Base‘𝑅)       ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝐴𝐵) → (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝐹𝐴) ∈ (SubGrp‘𝑆)))

Theoremgimcnv 17756 The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑆 GrpIso 𝑇) → 𝐹 ∈ (𝑇 GrpIso 𝑆))

Theoremgimco 17757 The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpIso 𝑈))

Theorembrgic 17758 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)

Theorembrgici 17759 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅𝑔 𝑆)

Theoremgicref 17760 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅 ∈ Grp → 𝑅𝑔 𝑅)

Theoremgiclcl 17761 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆𝑅 ∈ Grp)

Theoremgicrcl 17762 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅𝑔 𝑆𝑆 ∈ Grp)

Theoremgicsym 17763 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅𝑔 𝑆𝑆𝑔 𝑅)

Theoremgictr 17764 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
((𝑅𝑔 𝑆𝑆𝑔 𝑇) → 𝑅𝑔 𝑇)

Theoremgicer 17765 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)
𝑔 Er Grp

Theoremgicen 17766 Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝑅𝑔 𝑆𝐵𝐶)

Theoremgicsubgen 17767 A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

10.2.6  Group actions

Syntaxcga 17768 Extend class definition to include the class of group actions.
class GrpAct

Definitiondf-ga 17769* Define the class of all group actions. A group 𝐺 acts on a set 𝑆 if a permutation on 𝑆 is associated with every element of 𝐺 in such a way that the identity permutation on 𝑆 is associated with the neutral element of 𝐺, and the composition of the permutations associated with two elements of 𝐺 is identical with the permutation associated with the composition of these two elements (in the same order) in the group 𝐺. (Contributed by Jeff Hankins, 10-Aug-2009.)
GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ (Base‘𝑔) / 𝑏{𝑚 ∈ (𝑠𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})

Theoremisga 17770* The predicate "is a (left) group action." The group 𝐺 is said to act on the base set 𝑌 of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element 𝑔 of 𝐺 is a permutation of the elements of 𝑌 (see gapm 17785). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))

Theoremgagrp 17771 The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)
( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Theoremgaset 17772 The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)

Theoremgagrpid 17773 The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
0 = (0g𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)

Theoremgaf 17774 The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)       ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)

Theoremgafo 17775 A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)       ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)–onto𝑌)

Theoremgaass 17776 An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))

Theoremga0 17777 The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
(𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

Theoremgaid 17778 The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑆𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))

Theoremsubgga 17779* A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐻 = (𝐺s 𝑌)    &   𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))       (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Theoremgass 17780* A subset of a group action is a group action iff it is closed under the group action operation. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))

Theoremgasubg 17781 The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝐻 = (𝐺s 𝑆)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))

Theoremgaid2 17782* A group operation is a left group action of the group on itself. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐹 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥 + 𝑦))       (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpAct 𝑋))

Theoremgalcan 17783 The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))

Theoremgacan 17784 Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))

Theoremgapm 17785* The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐹 = (𝑥𝑌 ↦ (𝐴 𝑥))       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)

Theoremgaorb 17786* The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))

Theoremgaorber 17787* The orbit equivalence relation is an equivalence relation on the target set of the group action. (Contributed by NM, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   𝑋 = (Base‘𝐺)       ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)

Theoremgastacl 17788* The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐻 ∈ (SubGrp‘𝐺))

Theoremgastacos 17789* Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)       ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 𝐶 ↔ (𝐵 𝐴) = (𝐶 𝐴)))

Theoremorbstafun 17790* Existence and uniqueness for the function of orbsta 17792. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Fun 𝐹)

Theoremorbstaval 17791* Value of the function at a given equivalence class element. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)       ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝐵𝑋) → (𝐹‘[𝐵] ) = (𝐵 𝐴))

Theoremorbsta 17792* The Orbit-Stabilizer theorem. The mapping 𝐹 is a bijection from the cosets of the stabilizer subgroup of 𝐴 to the orbit of 𝐴. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)    &   𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)

Theoremorbsta2 17793* Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘[𝐴]𝑂) · (#‘𝐻)))

10.2.7  Centralizers and centers

Syntaxccntz 17794 Syntax for the centralizer of a set in a monoid.
class Cntz

Syntaxccntr 17795 Syntax for the centralizer of a monoid.
class Cntr

Definitiondf-cntz 17796* Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))

Definitiondf-cntr 17797 Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))

Theoremcntrval 17798 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑍𝐵) = (Cntr‘𝑀)

Theoremcntzfval 17799* First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))

Theoremcntzval 17800* Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >