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

Theoremodcl 18001 The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)

Theoremodf 18002 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       𝑂:𝑋⟶ℕ0

Theoremodid 18003 Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (𝐴𝑋 → ((𝑂𝐴) · 𝐴) = 0 )

Theoremodlem2 18004 Any positive annihilator of a group element is an upper bound on the (positive) order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐴𝑋𝑁 ∈ ℕ ∧ (𝑁 · 𝐴) = 0 ) → (𝑂𝐴) ∈ (1...𝑁))

Theoremodmodnn0 18005 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝑋𝑁 ∈ ℕ0) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑁 mod (𝑂𝐴)) · 𝐴) = (𝑁 · 𝐴))

Theoremmndodconglem 18006 Lemma for mndodcong 18007. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑀 < (𝑂𝐴))    &   (𝜑𝑁 < (𝑂𝐴))    &   (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴))       ((𝜑𝑀𝑁) → 𝑀 = 𝑁)

Theoremmndodcong 18007 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝑋) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑂𝐴) ∥ (𝑀𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))

Theoremmndodcongi 18008 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of 2 mod 10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑋 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝑂𝐴) ∥ (𝑀𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))

Theoremoddvdsnn0 18009 The only multiples of 𝐴 that are equal to the identity are the multiples of the order of 𝐴. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑋𝑁 ∈ ℕ0) → ((𝑂𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))

Theoremodnncl 18010 If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑁 ≠ 0 ∧ (𝑁 · 𝐴) = 0 )) → (𝑂𝐴) ∈ ℕ)

Theoremodmod 18011 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑁 mod (𝑂𝐴)) · 𝐴) = (𝑁 · 𝐴))

Theoremoddvds 18012 The only multiples of 𝐴 that are equal to the identity are the multiples of the order of 𝐴. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → ((𝑂𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))

Theoremoddvdsi 18013 Any group element is annihilated by any multiple of its order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∥ 𝑁) → (𝑁 · 𝐴) = 0 )

Theoremodcong 18014 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑂𝐴) ∥ (𝑀𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))

Theoremodeq 18015* The oddvds 18012 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℕ0) → (𝑁 = (𝑂𝐴) ↔ ∀𝑦 ∈ ℕ0 (𝑁𝑦 ↔ (𝑦 · 𝐴) = 0 )))

Theoremodval2 18016* A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = (𝑥 ∈ ℕ0𝑦 ∈ ℕ0 (𝑥𝑦 ↔ (𝑦 · 𝐴) = 0 )))

Theoremodmulgid 18017 A relationship between the order of a multiple and the order of the basepoint. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ 𝐾 ∈ ℤ) → ((𝑂‘(𝑁 · 𝐴)) ∥ 𝐾 ↔ (𝑂𝐴) ∥ (𝐾 · 𝑁)))

Theoremodmulg2 18018 The order of a multiple divides the order of the base point. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → (𝑂‘(𝑁 · 𝐴)) ∥ (𝑂𝐴))

Theoremodmulg 18019 Relationship between the order of an element and that of a multiple. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → (𝑂𝐴) = ((𝑁 gcd (𝑂𝐴)) · (𝑂‘(𝑁 · 𝐴))))

Theoremodmulgeq 18020 A multiple of a point of finite order only has the same order if the multiplier is relatively prime. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑂‘(𝑁 · 𝐴)) = (𝑂𝐴) ↔ (𝑁 gcd (𝑂𝐴)) = 1))

Theoremodbezout 18021* If 𝑁 is coprime to the order of 𝐴, there is a modular inverse 𝑥 to cancel multiplication by 𝑁. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑁 gcd (𝑂𝐴)) = 1) → ∃𝑥 ∈ ℤ (𝑥 · (𝑁 · 𝐴)) = 𝐴)

Theoremod1 18022 The order of the group identity is one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑂 = (od‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ Grp → (𝑂0 ) = 1)

Theoremodeq1 18023 The group identity is the unique element of a group with order one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑂 = (od‘𝐺)    &    0 = (0g𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) = 1 ↔ 𝐴 = 0 ))

Theoremodinv 18024 The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝐼 = (invg𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂‘(𝐼𝐴)) = (𝑂𝐴))

Theoremodf1 18025* The multiples of an element with infinite order form an infinite cyclic subgroup of 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) = 0 ↔ 𝐹:ℤ–1-1𝑋))

Theoremodinf 18026* The multiples of an element with infinite order form an infinite cyclic subgroup of 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)

Theoremdfod2 18027* An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = if(ran 𝐹 ∈ Fin, (#‘ran 𝐹), 0))

Theoremodcl2 18028 The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ)

Theoremoddvds2 18029 The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴𝑋) → (𝑂𝐴) ∥ (#‘𝑋))

Theoremsubmod 18030 The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
𝐻 = (𝐺s 𝑌)    &   𝑂 = (od‘𝐺)    &   𝑃 = (od‘𝐻)       ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))

Theoremsubgod 18031 The order of an element is the same in a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) (Proof shortened by Stefan O'Rear, 12-Sep-2015.)
𝐻 = (𝐺s 𝑌)    &   𝑂 = (od‘𝐺)    &   𝑃 = (od‘𝐻)       ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))

Theoremodsubdvds 18032 The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑂 = (od‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴𝑆) → (𝑂𝐴) ∥ (#‘𝑆))

Theoremodf1o1 18033* An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴}))

Theoremodf1o2 18034* An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∈ ℕ) → (𝑥 ∈ (0..^(𝑂𝐴)) ↦ (𝑥 · 𝐴)):(0..^(𝑂𝐴))–1-1-onto→(𝐾‘{𝐴}))

Theoremodhash 18035 An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) = 0) → (#‘(𝐾‘{𝐴})) = +∞)

Theoremodhash2 18036 If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∈ ℕ) → (#‘(𝐾‘{𝐴})) = (𝑂𝐴))

Theoremodhash3 18037 An element which generates a finite subgroup has order the size of that subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝐾‘{𝐴}) ∈ Fin) → (𝑂𝐴) = (#‘(𝐾‘{𝐴})))

Theoremodngen 18038* A cyclic subgroup of size (𝑂𝐴) has (ϕ‘(𝑂𝐴)) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∈ ℕ) → (#‘{𝑥 ∈ (𝐾‘{𝐴}) ∣ (𝑂𝑥) = (𝑂𝐴)}) = (ϕ‘(𝑂𝐴)))

Theoremgexval 18039* Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }       (𝐺𝑉𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))

Theoremgexlem1 18040* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥𝑋 (𝑦 · 𝑥) = 0 }       (𝐺𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸𝐼))

Theoremgexcl 18041 The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       (𝐺𝑉𝐸 ∈ ℕ0)

Theoremgexid 18042 Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (𝐴𝑋 → (𝐸 · 𝐴) = 0 )

Theoremgexlem2 18043* Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺𝑉𝑁 ∈ ℕ ∧ ∀𝑥𝑋 (𝑁 · 𝑥) = 0 ) → 𝐸 ∈ (1...𝑁))

Theoremgexdvdsi 18044 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝐸𝑁) → (𝑁 · 𝐴) = 0 )

Theoremgexdvds 18045* The only 𝑁 that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸𝑁 ↔ ∀𝑥𝑋 (𝑁 · 𝑥) = 0 ))

Theoremgexdvds2 18046* An integer divides the group exponent iff it divides all the group orders. In other words, the group exponent is the LCM of the orders of all the elements. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸𝑁 ↔ ∀𝑥𝑋 (𝑂𝑥) ∥ 𝑁))

Theoremgexod 18047 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) ∥ 𝐸)

Theoremgexcl3 18048* If the order of every group element is bounded by 𝑁, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ ∀𝑥𝑋 (𝑂𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ)

Theoremgexnnod 18049 Every group element has finite order if the exponent is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝐴𝑋) → (𝑂𝐴) ∈ ℕ)

Theoremgexcl2 18050 The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∈ ℕ)

Theoremgexdvds3 18051 The exponent of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∥ (#‘𝑋))

Theoremgex1 18052 A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1𝑜))

Theoremispgp 18053* A group is a 𝑃-group if every element has some power of 𝑃 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥𝑋𝑛 ∈ ℕ0 (𝑂𝑥) = (𝑃𝑛)))

Theorempgpprm 18054 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝑃 pGrp 𝐺𝑃 ∈ ℙ)

Theorempgpgrp 18055 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
(𝑃 pGrp 𝐺𝐺 ∈ Grp)

Theorempgpfi1 18056 A finite group with order a power of a prime 𝑃 is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → ((#‘𝑋) = (𝑃𝑁) → 𝑃 pGrp 𝐺))

Theorempgp0 18057 The identity subgroup is a 𝑃-group for every prime 𝑃. (Contributed by Mario Carneiro, 19-Jan-2015.)
0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺s { 0 }))

Theoremsubgpgp 18058 A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑆))

Theoremsylow1lem1 18059* Lemma for sylow1 18064. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (#‘𝑋) / (𝑃𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}       (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))

Theoremsylow1lem2 18060* Lemma for sylow1 18064. The function is a group action on 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       (𝜑 ∈ (𝐺 GrpAct 𝑆))

Theoremsylow1lem3 18061* Lemma for sylow1 18064. One of the orbits of the group action has p-adic valuation less than the prime count of the set 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑 → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))

Theoremsylow1lem4 18062* Lemma for sylow1 18064. The stabilizer subgroup of any element of 𝑆 is at most 𝑃𝑁 in size. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   (𝜑𝐵𝑆)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐵) = 𝐵}       (𝜑 → (#‘𝐻) ≤ (𝑃𝑁))

Theoremsylow1lem5 18063* Lemma for sylow1 18064. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   (𝜑𝐵𝑆)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐵) = 𝐵}    &   (𝜑 → (𝑃 pCnt (#‘[𝐵] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))       (𝜑 → ∃ ∈ (SubGrp‘𝐺)(#‘) = (𝑃𝑁))

Theoremsylow1 18064* Sylow's first theorem. If 𝑃𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))       (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(#‘𝑔) = (𝑃𝑁))

Theoremodcau 18065* Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 𝑃 contains an element of order 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (#‘𝑋)) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃)

Theorempgpfi 18066* The converse to pgpfi1 18056. A finite group is a 𝑃-group iff it has size some power of 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (#‘𝑋) = (𝑃𝑛))))

Theorempgpfi2 18067 Alternate version of pgpfi 18066. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (#‘𝑋) = (𝑃↑(𝑃 pCnt (#‘𝑋))))))

Theorempgphash 18068 The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)       ((𝑃 pGrp 𝐺𝑋 ∈ Fin) → (#‘𝑋) = (𝑃↑(𝑃 pCnt (#‘𝑋))))

Theoremisslw 18069* The property of being a Sylow subgroup. A Sylow 𝑃-subgroup is a 𝑃-group which has no proper supersets that are also 𝑃-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))

Theoremslwprm 18070 Reverse closure for the first argument of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)

Theoremslwsubg 18071 A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))

Theoremslwispgp 18072 Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐾)       ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))

Theoremslwpss 18073 A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐾)       ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ¬ 𝑃 pGrp 𝑆)

Theoremslwpgp 18074 A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐻)       (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)

Theorempgpssslw 18075* Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑆 = (𝐺s 𝐻)    &   𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↦ (#‘𝑥))       ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻𝑘)

Theoremslwn0 18076 Every finite group contains a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝑃 pSyl 𝐺) ≠ ∅)

Theoremsubgslw 18077 A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐻 = (𝐺s 𝑆)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

Theoremsylow2alem1 18078* Lemma for sylow2a 18080. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       ((𝜑𝐴𝑍) → [𝐴] = {𝐴})

Theoremsylow2alem2 18079* Lemma for sylow2a 18080. All the orbits which are not for fixed points have size 𝐺 ∣ / ∣ 𝐺𝑥 (where 𝐺𝑥 is the stabilizer subgroup) and thus are powers of 𝑃. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide 𝑃, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(#‘𝑧))

Theoremsylow2a 18080* A named lemma of Sylow's second and third theorems. If 𝐺 is a finite 𝑃-group that acts on the finite set 𝑌, then the set 𝑍 of all points of 𝑌 fixed by every element of 𝐺 has cardinality equivalent to the cardinality of 𝑌, mod 𝑃. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑𝑃 ∥ ((#‘𝑌) − (#‘𝑍)))

Theoremsylow2blem1 18081* Lemma for sylow2b 18084. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = [(𝐵 + 𝐶)] )

Theoremsylow2blem2 18082* Lemma for sylow2b 18084. Left multiplication in a subgroup 𝐻 is a group action on the set of all left cosets of 𝐾. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       (𝜑· ∈ ((𝐺s 𝐻) GrpAct (𝑋 / )))

Theoremsylow2blem3 18083* Sylow's second theorem. Putting together the results of sylow2a 18080 and the orbit-stabilizer theorem to show that 𝑃 does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some 𝑔𝑋 with 𝑔𝐾 = 𝑔𝐾 for all 𝐻. This implies that invg(𝑔)𝑔𝐾, so is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &   (𝜑𝑃 pGrp (𝐺s 𝐻))    &   (𝜑 → (#‘𝐾) = (𝑃↑(𝑃 pCnt (#‘𝑋))))    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))

Theoremsylow2b 18084* Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃𝑛 ∥ (#‘𝑋) with 𝑛 maximal. This is usually stated under the assumption that 𝐾 is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &   (𝜑𝑃 pGrp (𝐺s 𝐻))    &   (𝜑 → (#‘𝐾) = (𝑃↑(𝑃 pCnt (#‘𝑋))))    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))

Theoremslwhash 18085 A sylow subgroup has cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))       (𝜑 → (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))

Theoremfislw 18086 The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (#‘𝐻) = (𝑃↑(𝑃 pCnt (#‘𝑋))))))

Theoremsylow2 18087* Sylow's second theorem. See also sylow2b 18084 for the "hard" part of the proof. Any two Sylow 𝑃-subgroups are conjugate to one another, and hence the same size, namely 𝑃↑(𝑃 pCnt ∣ 𝑋 ∣ ) (see fislw 18086). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    + = (+g𝐺)    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 = ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))

Theoremsylow3lem1 18088* Lemma for sylow3 18094, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))       (𝜑 ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))

Theoremsylow3lem2 18089* Lemma for sylow3 18094, first part. The stabilizer of a given Sylow subgroup 𝐾 in the group action acting on all of 𝐺 is the normalizer NG(K). (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑𝐻 = 𝑁)

Theoremsylow3lem3 18090* Lemma for sylow3 18094, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup 𝐾. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑 → (#‘(𝑃 pSyl 𝐺)) = (#‘(𝑋 / (𝐺 ~QG 𝑁))))

Theoremsylow3lem4 18091* Lemma for sylow3 18094, first part. The number of Sylow subgroups is a divisor of the size of 𝐺 reduced by the size of a Sylow subgroup of 𝐺. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑 → (#‘(𝑃 pSyl 𝐺)) ∥ ((#‘𝑋) / (𝑃↑(𝑃 pCnt (#‘𝑋)))))

Theoremsylow3lem5 18092* Lemma for sylow3 18094, second part. Reduce the group action of sylow3lem1 18088 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    = (𝑥𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))       (𝜑 ∈ ((𝐺s 𝐾) GrpAct (𝑃 pSyl 𝐺)))

Theoremsylow3lem6 18093* Lemma for sylow3 18094, second part. Using the lemma sylow2a 18080, show that the number of sylow subgroups is equivalent mod 𝑃 to the number of fixed points under the group action. But 𝐾 is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so ((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    = (𝑥𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}       (𝜑 → ((#‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

Theoremsylow3 18094 Sylow's third theorem. The number of Sylow subgroups is a divisor of 𝐺 ∣ / 𝑑, where 𝑑 is the common order of a Sylow subgroup, and is equivalent to 1 mod 𝑃. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   𝑁 = (#‘(𝑃 pSyl 𝐺))       (𝜑 → (𝑁 ∥ ((#‘𝑋) / (𝑃↑(𝑃 pCnt (#‘𝑋)))) ∧ (𝑁 mod 𝑃) = 1))

10.2.11  Direct products

Syntaxclsm 18095 Extend class notation with subgroup sum.
class LSSum

Syntaxcpj1 18096 Extend class notation with left projection.
class proj1

Definitiondf-lsm 18097* Define subgroup sum (inner direct product of subgroups). (Contributed by NM, 28-Jan-2014.)
LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))

Definitiondf-pj1 18098* Define the left projection function, which takes two subgroups 𝑡, 𝑢 with trivial intersection and returns a function mapping the elements of the subgroup sum 𝑡 + 𝑢 to their projections onto 𝑡. (The other projection function can be obtained by swapping the roles of 𝑡 and 𝑢.) (Contributed by Mario Carneiro, 15-Oct-2015.)
proj1 = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑤)𝑦)))))

Theoremlsmfval 18099* The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))

Theoremlsmvalx 18100* Subspace sum value (for a group or vector space). Extended domain version of lsmval 18109. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))

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