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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubsubg 18001 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 18002 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 18003 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
0 = (0g𝐺)       (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))

Theoremcycsubgcl 18004* 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 18005* 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 18006* 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 18007* 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 18008* Weaken the condition of isnsg 18007 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 18009 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 18010 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
(𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))

Theoremnsgconj 18011 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 18012* 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 18013 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 18014 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 18015* 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 18016 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 18017* 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 18018* 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 18019* 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 18020* 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 18021* 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 18022* 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 18023 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
0 = (0g𝐺)       (𝐺 ∈ Grp → { 0 } ∈ (NrmSGrp‘𝐺))

Theoremnsgid 18024 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 18025 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑅 = (𝐺 ~QG 𝑆)       Rel 𝑅

Theoremeqgfval 18026* 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 18027 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 18028 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 18029* 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 18030 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 18031 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 18032 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 18033 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 18034 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 18035 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 18036 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 18037 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 18038 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 18039 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 18040 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  Elementary theory of group homomorphisms

Syntaxcghm 18041 Extend class notation with the generator of group hom-sets.
class GrpHom

Definitiondf-ghm 18042* 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 18043 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom GrpHom

Theoremisghm 18044* 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 18045* Property of a group homomorphism, similar to ismhm 17723. (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 18046 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 18047 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 18048 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 18049 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 18050 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 18051 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 18052 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 18053* 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 18054 A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremghmmhmb 18055 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 18056 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 18057 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 18058 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 18059 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 18060 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 18061 One direction of resghm2b 18062. (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 18062 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 18063 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 18064 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 18065 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 18066 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 18067 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 18068 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 18069 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 18070 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 18071 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 18072* 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 18073* 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 18074 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 18075* 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 18076* 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 18077* 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 18078* 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 18079* 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 18080* 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 18081* 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 18082* 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 18083 The class of group isomorphism sets.
class GrpIso

Syntaxcgic 18084 The class of the group isomorphism relation.
class 𝑔

Definitiondf-gim 18085* 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 18086 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 18087 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
GrpIso Fn (Grp × Grp)

Theoremisgim 18088 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 18089 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 18090 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 18091 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 21971. (Contributed by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑅)))

Theoremsubggim 18092 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 18093 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 18094 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 18095 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)

Theorembrgici 18096 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅𝑔 𝑆)

Theoremgicref 18097 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅 ∈ Grp → 𝑅𝑔 𝑅)

Theoremgiclcl 18098 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆𝑅 ∈ Grp)

Theoremgicrcl 18099 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅𝑔 𝑆𝑆 ∈ Grp)

Theoremgicsym 18100 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅𝑔 𝑆𝑆𝑔 𝑅)

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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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43661
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