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Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulgnn0ass 17401 Product of group multiples, generalized to 0. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgass 17402 Product of group multiples, generalized to . (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgassr 17403 Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋)))

Theoremmulgmodid 17404 Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋))

Theoremmulgsubdir 17405 Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑀𝑁) · 𝑋) = ((𝑀 · 𝑋) (𝑁 · 𝑋)))

Theoremmhmmulg 17406 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    × = (.g𝐻)       ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑁 ∈ ℕ0𝑋𝐵) → (𝐹‘(𝑁 · 𝑋)) = (𝑁 × (𝐹𝑋)))

Theoremmulgpropd 17407* Two structures with the same group-nature have the same group multiple function. 𝐾 is expected to either be V (when strong equality is available) or 𝐵 (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
· = (.g𝐺)    &    × = (.g𝐻)    &   (𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐵 = (Base‘𝐻))    &   (𝜑𝐵𝐾)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))       (𝜑· = × )

Theoremsubmmulgcl 17408 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
= (.g𝐺)       ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 𝑋) ∈ 𝑆)

Theoremsubmmulg 17409 A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
= (.g𝐺)    &   𝐻 = (𝐺s 𝑆)    &    · = (.g𝐻)       ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0𝑋𝑆) → (𝑁 𝑋) = (𝑁 · 𝑋))

Theorempwsmulg 17410 Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &    = (.g𝑌)    &    · = (.g𝑅)       (((𝑅 ∈ Mnd ∧ 𝐼𝑉) ∧ (𝑁 ∈ ℕ0𝑋𝐵𝐴𝐼)) → ((𝑁 𝑋)‘𝐴) = (𝑁 · (𝑋𝐴)))

10.2.3  Subgroups and Quotient groups

Syntaxcsubg 17411 Extend class notation with all subgroups of a group.
class SubGrp

Syntaxcnsg 17412 Extend class notation with all normal subgroups of a group.
class NrmSGrp

Syntaxcqg 17413 Quotient group equivalence class.
class ~QG

Definitiondf-subg 17414* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17432), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17427), contains the neutral element of the group (see subg0 17423) and contains the inverses for all of its elements (see subginvcl 17426). (Contributed by Mario Carneiro, 2-Dec-2014.)
SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})

Definitiondf-nsg 17415* Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g𝑤) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})

Definitiondf-eqg 17416* Define the equivalence relation in a quotient ring or quotient group (where 𝑖 is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})

Theoremissubg 17417 The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆𝐵 ∧ (𝐺s 𝑆) ∈ Grp))

Theoremsubgss 17418 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝐵)

Theoremsubgid 17419 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ∈ (SubGrp‘𝐺))

Theoremsubggrp 17420 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)

Theoremsubgbas 17421 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))

Theoremsubgrcl 17422 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
(𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)

Theoremsubg0 17423 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐻 = (𝐺s 𝑆)    &    0 = (0g𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g𝐻))

Theoremsubginv 17424 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐻 = (𝐺s 𝑆)    &   𝐼 = (invg𝐺)    &   𝐽 = (invg𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) = (𝐽𝑋))

Theoremsubg0cl 17425 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
0 = (0g𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 0𝑆)

Theoremsubginvcl 17426 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐼 = (invg𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆) → (𝐼𝑋) ∈ 𝑆)

Theoremsubgcl 17427 A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
+ = (+g𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Theoremsubgsubcl 17428 A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
= (-g𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 𝑌) ∈ 𝑆)

Theoremsubgsub 17429 The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
= (-g𝐺)    &   𝐻 = (𝐺s 𝑆)    &   𝑁 = (-g𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 𝑌) = (𝑋𝑁𝑌))

Theoremsubgmulgcl 17430 Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
· = (.g𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) ∈ 𝑆)

Theoremsubgmulg 17431 A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
· = (.g𝐺)    &   𝐻 = (𝐺s 𝑆)    &    = (.g𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = (𝑁 𝑋))

Theoremissubg2 17432* Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵𝑆 ≠ ∅ ∧ ∀𝑥𝑆 (∀𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ∧ (𝐼𝑥) ∈ 𝑆))))

Theoremissubgrpd2 17433* Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑𝐼 ∈ Grp)       (𝜑𝐷 ∈ (SubGrp‘𝐼))

Theoremissubgrpd 17434* Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑𝐼 ∈ Grp)       (𝜑𝑆 ∈ Grp)

Theoremissubg3 17435* A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐼 = (invg𝐺)       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑥𝑆 (𝐼𝑥) ∈ 𝑆)))

Theoremissubg4 17436* A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆𝐵𝑆 ≠ ∅ ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 𝑦) ∈ 𝑆)))

Theoremgrpissubg 17437 If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)       ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubGrp‘𝐺)))

Theoremresgrpisgrp 17438 If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)       ((𝐺 ∈ Grp ∧ 𝐻 ∈ Grp) → ((𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → (𝐺s 𝑆) ∈ Grp))

Theoremsubgsubm 17439 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))

Theoremsubsubg 17440 A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆)))

Theoremsubgint 17441 The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
((𝑆 ⊆ (SubGrp‘𝐺) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubGrp‘𝐺))

Theorem0subg 17442 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
0 = (0g𝐺)       (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))

Theoremcycsubgcl 17443* The set of integer powers of an element 𝐴 of a group forms a subgroup containing 𝐴, called the cyclic group generated by the element 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹))

Theoremcycsubgss 17444* The cyclic subgroup generated by an element 𝐴 is a subset of any subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑆) → ran 𝐹𝑆)

Theoremcycsubg 17445* The cyclic group generated by 𝐴 is the smallest subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ran 𝐹 = {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴𝑠})

Theoremisnsg 17446* Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))

Theoremisnsg2 17447* Weaken the condition of isnsg 17446 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))

Theoremnsgbi 17448 Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))

Theoremnsgsubg 17449 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
(𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))

Theoremnsgconj 17450 The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) ∈ 𝑆)

Theoremisnsg3 17451* A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))

Theoremsubgacs 17452 Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))

Theoremnsgacs 17453 Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → (NrmSGrp‘𝐺) ∈ (ACS‘𝐵))

Theoremcycsubg2 17454* The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐾‘{𝐴}) = ran 𝐹)

Theoremcycsubg2cl 17455 Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴}))

Theoremelnmz 17456* Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}       (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑧𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))

Theoremnmzbi 17457* Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}       ((𝐴𝑁𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))

Theoremnmzsubg 17458* The normalizer NG(S) of a subset 𝑆 of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))

Theoremssnmz 17459* A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)

Theoremisnsg4 17460* A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 = 𝑋))

Theoremnmznsg 17461* Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐻 = (𝐺s 𝑁)       (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))

Theorem0nsg 17462 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
0 = (0g𝐺)       (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺))

Theoremnsgid 17463 The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺))

Theoremreleqg 17464 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑅 = (𝐺 ~QG 𝑆)       Rel 𝑅

Theoremeqgfval 17465* Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (invg𝐺)    &    + = (+g𝐺)    &   𝑅 = (𝐺 ~QG 𝑆)       ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})

Theoremeqgval 17466 Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (invg𝐺)    &    + = (+g𝐺)    &   𝑅 = (𝐺 ~QG 𝑆)       ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))

Theoremeqger 17467 The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    = (𝐺 ~QG 𝑌)       (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)

Theoremeqglact 17468* A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑋 = (Base‘𝐺)    &    = (𝐺 ~QG 𝑌)    &    + = (+g𝐺)       ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Theoremeqgid 17469 The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑋 = (Base‘𝐺)    &    = (𝐺 ~QG 𝑌)    &    0 = (0g𝐺)       (𝑌 ∈ (SubGrp‘𝐺) → [ 0 ] = 𝑌)

Theoremeqgen 17470 Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑋 = (Base‘𝐺)    &    = (𝐺 ~QG 𝑌)       ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / )) → 𝑌𝐴)

Theoremeqgcpbl 17471 The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝐺)    &    = (𝐺 ~QG 𝑌)    &    + = (+g𝐺)       (𝑌 ∈ (NrmSGrp‘𝐺) → ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷)))

Theoremqusgrp 17472 If 𝑌 is a normal subgroup of 𝐺, then 𝐻 = 𝐺 / 𝑌 is a group, called the quotient of 𝐺 by 𝑌. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))       (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)

Theoremquseccl 17473 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵)

Theoremqusadd 17474 Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (+g𝐻)       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋𝑉𝑌𝑉) → ([𝑋](𝐺 ~QG 𝑆) [𝑌](𝐺 ~QG 𝑆)) = [(𝑋 + 𝑌)](𝐺 ~QG 𝑆))

Theoremqus0 17475 Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &    0 = (0g𝐺)       (𝑆 ∈ (NrmSGrp‘𝐺) → [ 0 ](𝐺 ~QG 𝑆) = (0g𝐻))

Theoremqusinv 17476 Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &   𝐼 = (invg𝐺)    &   𝑁 = (invg𝐻)       ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋𝑉) → (𝑁‘[𝑋](𝐺 ~QG 𝑆)) = [(𝐼𝑋)](𝐺 ~QG 𝑆))

Theoremqussub 17477 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 17478 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 17479 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 17480 Extend class notation with the generator of group hom-sets.
class GrpHom

Definitiondf-ghm 17481* 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 17482 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom GrpHom

Theoremisghm 17483* 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 17484* Property of a group homomorphism, similar to ismhm 17160. (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 17485 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 17486 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 17487 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 17488 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 17489 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 17490 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 17491 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 17492* 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 17493 A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremghmmhmb 17494 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 17495 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 17496 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 17497 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 17498 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 17499 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 17500 One direction of resghm2b 17501. (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))

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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 42301-42360
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