![]() |
Metamath
Proof Explorer Theorem List (p. 197 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | submod 19601 | The order of an element is the same in a submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.) |
⊢ 𝐻 = (𝐺 ↾s 𝑌) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑃 = (od‘𝐻) ⇒ ⊢ ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ 𝑌) → (𝑂‘𝐴) = (𝑃‘𝐴)) | ||
Theorem | subgod 19602 | 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 = (𝐺 ↾s 𝑌) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝑃 = (od‘𝐻) ⇒ ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑌) → (𝑂‘𝐴) = (𝑃‘𝐴)) | ||
Theorem | odsubdvds 19603 | 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 ∧ 𝐴 ∈ 𝑆) → (𝑂‘𝐴) ∥ (♯‘𝑆)) | ||
Theorem | odf1o1 19604* | An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴})) | ||
Theorem | odf1o2 19605* | An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 ∈ (0..^(𝑂‘𝐴)) ↦ (𝑥 · 𝐴)):(0..^(𝑂‘𝐴))–1-1-onto→(𝐾‘{𝐴})) | ||
Theorem | odhash 19606 | 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 ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (♯‘(𝐾‘{𝐴})) = +∞) | ||
Theorem | odhash2 19607 | 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 ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(𝐾‘{𝐴})) = (𝑂‘𝐴)) | ||
Theorem | odhash3 19608 | 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 ∧ 𝐴 ∈ 𝑋 ∧ (𝐾‘{𝐴}) ∈ Fin) → (𝑂‘𝐴) = (♯‘(𝐾‘{𝐴}))) | ||
Theorem | odngen 19609* | A cyclic subgroup of size (𝑂‘𝐴) has (ϕ‘(𝑂‘𝐴)) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘{𝑥 ∈ (𝐾‘{𝐴}) ∣ (𝑂‘𝑥) = (𝑂‘𝐴)}) = (ϕ‘(𝑂‘𝐴))) | ||
Theorem | gexval 19610* | Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) | ||
Theorem | gexlem1 19611* | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ⇒ ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) | ||
Theorem | gexcl 19612 | The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0) | ||
Theorem | gexid 19613 | Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (gEx‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 ) | ||
Theorem | gexlem2 19614* | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ) → 𝐸 ∈ (1...𝑁)) | ||
Theorem | gexdvdsi 19615 | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∥ 𝑁) → (𝑁 · 𝐴) = 0 ) | ||
Theorem | gexdvds 19616* | 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) & ⊢ · = (.g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 )) | ||
Theorem | gexdvds2 19617* | 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) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ 𝑁)) | ||
Theorem | gexod 19618 | 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 ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸) | ||
Theorem | gexcl3 19619* | 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 ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ) | ||
Theorem | gexnnod 19620 | 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 ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ) | ||
Theorem | gexcl2 19621 | 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) | ||
Theorem | gexdvds3 19622 | 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)) | ||
Theorem | gex1 19623 | 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)) | ||
Theorem | ispgp 19624* | 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 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛))) | ||
Theorem | pgpprm 19625 | Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ) | ||
Theorem | pgpgrp 19626 | Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp) | ||
Theorem | pgpfi1 19627 | 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) = (𝑃↑𝑁) → 𝑃 pGrp 𝐺)) | ||
Theorem | pgp0 19628 | The identity subgroup is a 𝑃-group for every prime 𝑃. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺 ↾s { 0 })) | ||
Theorem | subgpgp 19629 | A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑆)) | ||
Theorem | sylow1lem1 19630* | Lemma for sylow1 19635. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (♯‘𝑋) / (𝑃↑𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ⇒ ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) | ||
Theorem | sylow1lem2 19631* | Lemma for sylow1 19635. The function ⊕ is a group action on 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ⇒ ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆)) | ||
Theorem | sylow1lem3 19632* | Lemma for sylow1 19635. 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) | ||
Theorem | sylow1lem4 19633* | Lemma for sylow1 19635. The stabilizer subgroup of any element of 𝑆 is at most 𝑃↑𝑁 in size. (Contributed by Mario Carneiro, 15-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵} ⇒ ⊢ (𝜑 → (♯‘𝐻) ≤ (𝑃↑𝑁)) | ||
Theorem | sylow1lem5 19634* | Lemma for sylow1 19635. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃↑𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) & ⊢ + = (+g‘𝐺) & ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵} & ⊢ (𝜑 → (𝑃 pCnt (♯‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ⇒ ⊢ (𝜑 → ∃ℎ ∈ (SubGrp‘𝐺)(♯‘ℎ) = (𝑃↑𝑁)) | ||
Theorem | sylow1 19635* | Sylow's first theorem. If 𝑃↑𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃↑𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋)) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁)) | ||
Theorem | odcau 19636* | 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)) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃) | ||
Theorem | pgpfi 19637* | The converse to pgpfi1 19627. 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) = (𝑃↑𝑛)))) | ||
Theorem | pgpfi2 19638 | Alternate version of pgpfi 19637. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))))) | ||
Theorem | pgphash 19639 | The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.) |
⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) | ||
Theorem | isslw 19640* | 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)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘))) | ||
Theorem | slwprm 19641 | 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) | ||
Theorem | slwsubg 19642 | A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺)) | ||
Theorem | slwispgp 19643 | Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑆 = (𝐺 ↾s 𝐾) ⇒ ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)) | ||
Theorem | slwpss 19644 | A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑆 = (𝐺 ↾s 𝐾) ⇒ ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆) | ||
Theorem | slwpgp 19645 | A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑆 = (𝐺 ↾s 𝐻) ⇒ ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆) | ||
Theorem | pgpssslw 19646* | Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑆 = (𝐺 ↾s 𝐻) & ⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↦ (♯‘𝑥)) ⇒ ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻 ⊆ 𝑘) | ||
Theorem | slwn0 19647 | 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) ≠ ∅) | ||
Theorem | subgslw 19648 | 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 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻)) | ||
Theorem | sylow2alem1 19649* | Lemma for sylow2a 19651. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌)) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴}) | ||
Theorem | sylow2alem2 19650* | Lemma for sylow2a 19651. 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)) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) | ||
Theorem | sylow2a 19651* | 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)) & ⊢ (𝜑 → 𝑃 pGrp 𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} & ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} ⇒ ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍))) | ||
Theorem | sylow2blem1 19652* | Lemma for sylow2b 19655. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝐾) & ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ ) | ||
Theorem | sylow2blem2 19653* | Lemma for sylow2b 19655. 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) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝐾) & ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) ⇒ ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ ))) | ||
Theorem | sylow2blem3 19654* | Sylow's second theorem. Putting together the results of sylow2a 19651 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 𝑔 ∈ 𝑋 with ℎ𝑔𝐾 = 𝑔𝐾 for all ℎ ∈ 𝐻. This implies that invg(𝑔)ℎ𝑔 ∈ 𝐾, so ℎ is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ ∼ = (𝐺 ~QG 𝐾) & ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) & ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) & ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | ||
Theorem | sylow2b 19655* | Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃↑𝑛 ∥ (♯‘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) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻)) & ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | ||
Theorem | slwhash 19656 | 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) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) ⇒ ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋)))) | ||
Theorem | fislw 19657 | 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)))))) | ||
Theorem | sylow2 19658* | Sylow's second theorem. See also sylow2b 19655 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 19657). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))) | ||
Theorem | sylow3lem1 19659* | Lemma for sylow3 19665, first part. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ⇒ ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺))) | ||
Theorem | sylow3lem2 19660* | Lemma for sylow3 19665, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} ⇒ ⊢ (𝜑 → 𝐻 = 𝑁) | ||
Theorem | sylow3lem3 19661* | Lemma for sylow3 19665, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} ⇒ ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~QG 𝑁)))) | ||
Theorem | sylow3lem4 19662* | Lemma for sylow3 19665, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾} & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)} ⇒ ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋))))) | ||
Theorem | sylow3lem5 19663* | Lemma for sylow3 19665, second part. Reduce the group action of sylow3lem1 19659 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) ⇒ ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺))) | ||
Theorem | sylow3lem6 19664* | Lemma for sylow3 19665, second part. Using the lemma sylow2a 19651, 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ + = (+g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺)) & ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥))) & ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ⇒ ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1) | ||
Theorem | sylow3 19665 | 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) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ 𝑁 = (♯‘(𝑃 pSyl 𝐺)) ⇒ ⊢ (𝜑 → (𝑁 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ∧ (𝑁 mod 𝑃) = 1)) | ||
Syntax | clsm 19666 | Extend class notation with subgroup sum. |
class LSSum | ||
Syntax | cpj1 19667 | Extend class notation with left projection. |
class proj1 | ||
Definition | df-lsm 19668* | 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 (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)))) | ||
Definition | df-pj1 19669* | 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) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑤)𝑦))))) | ||
Theorem | lsmfval 19670* | 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) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) | ||
Theorem | lsmvalx 19671* | Subspace sum value (for a group or vector space). Extended domain version of lsmval 19680. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) | ||
Theorem | lsmelvalx 19672* | Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19681. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) | ||
Theorem | lsmelvalix 19673 | Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) | ||
Theorem | oppglsm 19674 | The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.) |
⊢ 𝑂 = (oppg‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇) | ||
Theorem | lsmssv 19675 | Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) ⊆ 𝐵) | ||
Theorem | lsmless1x 19676 | Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑅 ⊆ 𝑇) → (𝑅 ⊕ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmless2x 19677 | Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈)) | ||
Theorem | lsmub1x 19678 | Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ⊆ 𝐵 ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmub2x 19679 | Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmval 19680* | Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) | ||
Theorem | lsmelval 19681* | Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) | ||
Theorem | lsmelvali 19682 | Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmelvalm 19683* | Subgroup sum membership analogue of lsmelval 19681 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ − = (-g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) | ||
Theorem | lsmelvalmi 19684 | Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ − = (-g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝑇) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmsubm 19685 | The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺)) | ||
Theorem | lsmsubg 19686 | The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺)) | ||
Theorem | lsmcom2 19687 | Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) | ||
Theorem | smndlsmidm 19688 | The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmub1 19689 | Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmub2 19690 | Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmunss 19691 | Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ∪ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmless1 19692 | Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑇) → (𝑆 ⊕ 𝑈) ⊆ (𝑇 ⊕ 𝑈)) | ||
Theorem | lsmless2 19693 | Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) | ||
Theorem | lsmless12 19694 | Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈)) | ||
Theorem | lsmidm 19695 | Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmlub 19696 | The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈) ↔ (𝑆 ⊕ 𝑇) ⊆ 𝑈)) | ||
Theorem | lsmss1 19697 | Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑈) → (𝑇 ⊕ 𝑈) = 𝑈) | ||
Theorem | lsmss1b 19698 | Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ 𝑈 ↔ (𝑇 ⊕ 𝑈) = 𝑈)) | ||
Theorem | lsmss2 19699 | Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑈 ⊆ 𝑇) → (𝑇 ⊕ 𝑈) = 𝑇) | ||
Theorem | lsmss2b 19700 | Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |