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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisnsg3 18301* 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 18302 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 18303 Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → (NrmSGrp‘𝐺) ∈ (ACS‘𝐵))

Theoremelnmz 18304* 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 18305* 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 18306* 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 18307* 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 18308* 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 18309* 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 18310 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
0 = (0g𝐺)       (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺))

Theoremnsgid 18311 The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → 𝐵 ∈ (NrmSGrp‘𝐺))

Theorem0idnsgd 18312 The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)       (𝜑 → {{ 0 }, 𝐵} ⊆ (NrmSGrp‘𝐺))

Theoremtrivnsgd 18313 The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐵 = { 0 })       (𝜑 → (NrmSGrp‘𝐺) = {𝐵})

Theoremtriv1nsgd 18314 A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐵 = { 0 })       (𝜑 → (NrmSGrp‘𝐺) ≈ 1o)

Theorem1nsgtrivd 18315 A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑 → (NrmSGrp‘𝐺) ≈ 1o)       (𝜑𝐵 = { 0 })

Theoremreleqg 18316 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑅 = (𝐺 ~QG 𝑆)       Rel 𝑅

Theoremeqgfval 18317* 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 18318 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 18319 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 18320* 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 18321 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 18322 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 18323 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 18324 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 18325 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 18326 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 18327 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 18328 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 18329 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 18330 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 18331 Lagrange's 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  Cyclic monoids and groups

This section contains some preliminary results about cyclic monoids and groups before the class CycGrp of cyclic groups (see df-cyg 18986) is defined in the context of Abelian groups.

Theoremcycsubmel 18332* Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       (𝑋𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))

Theoremcycsubmcl 18333* The set of nonnegative integer powers of an element 𝐴 contains 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       (𝐴𝐵𝐴𝐶)

Theoremcycsubm 18334* The set of nonnegative integer powers of an element 𝐴 of a monoid forms a submonoid containing 𝐴 (see cycsubmcl 18333), called the cyclic monoid generated by the element 𝐴. This corresponds to the statement in [Lang] p. 6. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       ((𝐺 ∈ Mnd ∧ 𝐴𝐵) → 𝐶 ∈ (SubMnd‘𝐺))

Theoremcyccom 18335* Condition for an operation to be commutative. Lemma for cycsubmcom 18336 and cygabl 18999. Formerly part of proof for cygabl 18999. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 20-Jan-2024.)
(𝜑 → ∀𝑐𝐶𝑥𝑍 𝑐 = (𝑥 · 𝐴))    &   (𝜑 → ∀𝑚𝑍𝑛𝑍 ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴)))    &   (𝜑𝑋𝐶)    &   (𝜑𝑌𝐶)    &   (𝜑𝑍 ⊆ ℂ)       (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremcycsubmcom 18336* The operation of a monoid is commutative over the set of nonnegative integer powers of an element 𝐴 of the monoid. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    &    + = (+g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝐵) ∧ (𝑋𝐶𝑌𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremcycsubggend 18337* The cyclic subgroup generated by 𝐴 includes its generator. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ ran 𝐹)

Theoremcycsubgcl 18338* 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 18339* 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 18340* 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})

Theoremcycsubgcld 18341* The cyclic subgroup generated by 𝐴 is a subgroup. Deduction related to cycsubgcl 18338. (Contributed by Rohan Ridenour, 3-Aug-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 𝐴))    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐴𝐵)       (𝜑 → ran 𝐹 ∈ (SubGrp‘𝐺))

Theoremcycsubg2 18342* 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 18343 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}))

10.2.5  Elementary theory of group homomorphisms

Syntaxcghm 18344 Extend class notation with the generator of group hom-sets.
class GrpHom

Definitiondf-ghm 18345* 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 18346 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom GrpHom

Theoremisghm 18347* 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 18348* Property of a group homomorphism, similar to ismhm 17947. (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 18349 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 18350 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 18351 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 18352 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 18353 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 18354 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 18355 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 18356* 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 18357 A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremghmmhmb 18358 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 18359 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 18360 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 18361 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 18362 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 18363 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 18364 One direction of resghm2b 18365. (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 18365 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 18366 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 18367 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 18368 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 18369 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 18370 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 18371 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 18372 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 18373 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 18374 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 18375* 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 18376* 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 18377 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 18378* 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 18379* 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 18380* 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 18381* 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 18382* 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 18383* 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 18384* 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 18385* 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))

10.2.6  Isomorphisms of groups

Syntaxcgim 18386 The class of group isomorphism sets.
class GrpIso

Syntaxcgic 18387 The class of the group isomorphism relation.
class 𝑔

Definitiondf-gim 18388* 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 18389 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 18390 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
GrpIso Fn (Grp × Grp)

Theoremisgim 18391 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 18392 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 18393 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 18394 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 22353. (Contributed by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))

Theoremsubggim 18395 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 18396 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 18397 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 18398 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)

Theorembrgici 18399 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅𝑔 𝑆)

Theoremgicref 18400 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅 ∈ Grp → 𝑅𝑔 𝑅)

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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 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