Order of an element #
This file defines the order of an element of a finite group. For a finite group G the order of
x ∈ G is the minimal n ≥ 1 such that x ^ n = 1.
Main definitions #
is_of_fin_orderis a predicate on an elementxof a monoidGsaying thatxis of finite order.is_of_fin_add_orderis the additive analogue ofis_of_find_order.order_of xdefines the order of an elementxof a monoidG, by convention its value is0ifxhas infinite order.add_order_ofis the additive analogue oforder_of.
Tags #
order of an element
theorem
is_periodic_pt_add_iff_nsmul_eq_zero
{G : Type u}
{n : ℕ}
[add_monoid G]
(x : G) :
function.is_periodic_pt (has_add.add x) n 0 ↔ n • x = 0
theorem
is_periodic_pt_mul_iff_pow_eq_one
{G : Type u}
{n : ℕ}
[monoid G]
(x : G) :
function.is_periodic_pt (has_mul.mul x) n 1 ↔ x ^ n = 1
is_of_fin_add_order is a predicate on an element a of an additive monoid to be of finite
order, i.e. there exists n ≥ 1 such that n • a = 0.
Equations
- is_of_fin_add_order a = (0 ∈ function.periodic_pts (has_add.add a))
is_of_fin_order is a predicate on an element x of a monoid to be of finite order, i.e. there
exists n ≥ 1 such that x ^ n = 1.
Equations
- is_of_fin_order x = (1 ∈ function.periodic_pts (has_mul.mul x))
order_of x is the order of the element x, i.e. the n ≥ 1, s.t. x ^ n = 1 if it exists.
Otherwise, i.e. if x is of infinite order, then order_of x is 0 by convention.
Equations
- order_of x = function.minimal_period (has_mul.mul x) 1
add_order_of a is the order of the element a, i.e. the n ≥ 1, s.t. n • a = 0 if it
exists. Otherwise, i.e. if a is of infinite order, then add_order_of a is 0 by convention.
Equations
@[simp]
@[simp]
theorem
add_order_of_pos'
{G : Type u}
{x : G}
[add_monoid G]
(h : is_of_fin_add_order x) :
0 < add_order_of x
theorem
add_order_of_eq_zero
{G : Type u}
{x : G}
[add_monoid G]
(h : ¬is_of_fin_add_order x) :
add_order_of x = 0
theorem
add_order_of_eq_zero_iff
{G : Type u}
{x : G}
[add_monoid G] :
add_order_of x = 0 ↔ ¬is_of_fin_add_order x
theorem
nsmul_ne_zero_of_lt_add_order_of'
{G : Type u}
{x : G}
{n : ℕ}
[add_monoid G]
(n0 : n ≠ 0)
(h : n < add_order_of x) :
theorem
add_order_of_le_of_nsmul_eq_zero
{G : Type u}
{x : G}
{n : ℕ}
[add_monoid G]
(hn : 0 < n)
(h : n • x = 0) :
add_order_of x ≤ n
@[simp]
theorem
add_monoid.order_of_eq_one_iff
{G : Type u}
{x : G}
[add_monoid G] :
add_order_of x = 1 ↔ x = 0
theorem
nsmul_eq_mod_add_order_of
{G : Type u}
{x : G}
[add_monoid G]
{n : ℕ} :
n • x = (n % add_order_of x) • x
theorem
add_order_of_dvd_of_nsmul_eq_zero
{G : Type u}
{x : G}
{n : ℕ}
[add_monoid G]
(h : n • x = 0) :
add_order_of x ∣ n
theorem
add_order_of_dvd_iff_nsmul_eq_zero
{G : Type u}
{x : G}
[add_monoid G]
{n : ℕ} :
add_order_of x ∣ n ↔ n • x = 0
theorem
exists_nsmul_eq_self_of_coprime
{G : Type u}
{x : G}
{n : ℕ}
[add_monoid G]
(h : n.coprime (add_order_of x)) :
theorem
add_order_of_eq_of_nsmul_and_div_prime_nsmul
{G : Type u}
{x : G}
{n : ℕ}
[add_monoid G]
(hn : 0 < n)
(hx : n • x = 0)
(hd : ∀ (p : ℕ), nat.prime p → p ∣ n → (n / p) • x ≠ 0) :
add_order_of x = n
theorem
add_order_of_eq_add_order_of_iff
{G : Type u}
{x : G}
[add_monoid G]
{H : Type u_1}
[add_monoid H]
{y : H} :
add_order_of x = add_order_of y ↔ ∀ (n : ℕ), n • x = 0 ↔ n • y = 0
theorem
add_order_of_injective
{G : Type u}
[add_monoid G]
{H : Type u_1}
[add_monoid H]
(f : G →+ H)
(hf : function.injective ⇑f)
(x : G) :
add_order_of (⇑f x) = add_order_of x
@[simp, norm_cast]
theorem
add_order_of_nsmul'
{G : Type u}
(x : G)
{n : ℕ}
[add_monoid G]
(h : n ≠ 0) :
add_order_of (n • x) = add_order_of x / (add_order_of x).gcd n
theorem
add_order_of_nsmul''
{G : Type u}
(x : G)
(n : ℕ)
[add_monoid G]
(h : is_of_fin_add_order x) :
add_order_of (n • x) = add_order_of x / (add_order_of x).gcd n
theorem
add_commute.order_of_add_dvd_lcm
{G : Type u}
[add_monoid G]
{x y : G}
(h : add_commute x y) :
add_order_of (x + y) ∣ (add_order_of x).lcm (add_order_of y)
theorem
add_commute.add_order_of_add_dvd_mul_add_order_of
{G : Type u}
[add_monoid G]
{x y : G}
(h : add_commute x y) :
add_order_of (x + y) ∣ (add_order_of x) * add_order_of y
theorem
add_commute.add_order_of_add_eq_mul_add_order_of_of_coprime
{G : Type u}
[add_monoid G]
{x y : G}
(h : add_commute x y)
(hco : (add_order_of x).coprime (add_order_of y)) :
add_order_of (x + y) = (add_order_of x) * add_order_of y
theorem
add_order_of_eq_prime
{G : Type u}
{x : G}
[add_monoid G]
{p : ℕ}
[hp : fact (nat.prime p)]
(hg : p • x = 0)
(hg1 : x ≠ 0) :
add_order_of x = p
theorem
nsmul_injective_aux
{G : Type u}
(x : G)
{n m : ℕ}
[add_left_cancel_monoid G]
(h : n ≤ m)
(hm : m < add_order_of x)
(eq : n • x = m • x) :
n = m
theorem
nsmul_injective_of_lt_add_order_of
{G : Type u}
(x : G)
{n m : ℕ}
[add_left_cancel_monoid G]
(hn : n < add_order_of x)
(hm : m < add_order_of x)
(eq : n • x = m • x) :
n = m
theorem
mem_powers_iff_mem_range_order_of'
{G : Type u}
(x y : G)
[left_cancel_monoid G]
[decidable_eq G]
(hx : 0 < order_of x) :
y ∈ submonoid.powers x ↔ y ∈ finset.image (pow x) (finset.range (order_of x))
theorem
mem_multiples_iff_mem_range_add_order_of'
{G : Type u}
(x y : G)
[add_left_cancel_monoid G]
[decidable_eq G]
(hx : 0 < add_order_of x) :
y ∈ add_submonoid.multiples x ↔ y ∈ finset.image (λ (ᾰ : ℕ), ᾰ • x) (finset.range (add_order_of x))
@[simp, norm_cast]
theorem
nsmul_inj_mod
{G : Type u}
{x : G}
[add_group G]
{n m : ℕ} :
n • x = m • x ↔ n % add_order_of x = m % add_order_of x
theorem
sum_card_order_of_eq_card_pow_eq_one
{G : Type u}
{n : ℕ}
[fintype G]
[monoid G]
[decidable_eq G]
(hn : 0 < n) :
∑ (m : ℕ) in finset.filter (λ (_x : ℕ), _x ∣ n) (finset.range n.succ), (finset.filter (λ (x : G), order_of x = m) finset.univ).card = (finset.filter (λ (x : G), x ^ n = 1) finset.univ).card
theorem
sum_card_add_order_of_eq_card_nsmul_eq_zero
{G : Type u}
{n : ℕ}
[fintype G]
[add_monoid G]
[decidable_eq G]
(hn : 0 < n) :
∑ (m : ℕ) in finset.filter (λ (_x : ℕ), _x ∣ n) (finset.range n.succ), (finset.filter (λ (x : G), add_order_of x = m) finset.univ).card = (finset.filter (λ (x : G), n • x = 0) finset.univ).card
theorem
order_of_le_card_univ
{G : Type u}
{x : G}
[fintype G]
[left_cancel_monoid G] :
order_of x ≤ fintype.card G
This is the same as `order_of_pos' but with one fewer explicit assumption since this is automatic in case of a finite cancellative monoid.
theorem
add_order_of_pos
{G : Type u}
[fintype G]
[add_left_cancel_monoid G]
(x : G) :
0 < add_order_of x
This is the same as `add_order_of_pos' but with one fewer explicit assumption since this is automatic in case of a finite cancellative additive monoid.
theorem
add_order_of_nsmul
{G : Type u}
{n : ℕ}
[fintype G]
[add_left_cancel_monoid G]
(x : G) :
add_order_of (n • x) = add_order_of x / (add_order_of x).gcd n
This is the same as add_order_of_nsmul' and add_order_of_nsmul but with one assumption less
which is automatic in the case of a finite cancellative additive monoid.
theorem
mem_multiples_iff_mem_range_add_order_of
{G : Type u}
{x y : G}
[fintype G]
[add_left_cancel_monoid G]
[decidable_eq G] :
y ∈ add_submonoid.multiples x ↔ y ∈ finset.image (λ (ᾰ : ℕ), ᾰ • x) (finset.range (add_order_of x))
theorem
mem_powers_iff_mem_range_order_of
{G : Type u}
{x y : G}
[fintype G]
[left_cancel_monoid G]
[decidable_eq G] :
y ∈ submonoid.powers x ↔ y ∈ finset.image (pow x) (finset.range (order_of x))
@[protected, instance]
noncomputable
def
decidable_multiples
{G : Type u}
{x : G}
[fintype G]
[add_left_cancel_monoid G]
[decidable_eq G] :
decidable_pred (λ (_x : G), _x ∈ add_submonoid.multiples x)
Equations
- decidable_multiples = id (λ (y : G), decidable_of_iff' (y ∈ finset.image (λ (ᾰ : ℕ), ᾰ • x) (finset.range (add_order_of x))) _)
@[protected, instance]
noncomputable
def
decidable_powers
{G : Type u}
{x : G}
[fintype G]
[left_cancel_monoid G]
[decidable_eq G] :
decidable_pred (λ (_x : G), _x ∈ submonoid.powers x)
Equations
- decidable_powers = id (λ (y : G), decidable_of_iff' (y ∈ finset.image (pow x) (finset.range (order_of x))) _)
noncomputable
def
fin_equiv_multiples
{G : Type u}
[fintype G]
[add_left_cancel_monoid G]
(x : G) :
fin (add_order_of x) ≃ ↥↑(add_submonoid.multiples x)
The equivalence between fin (add_order_of a) and
add_submonoid.multiples a, sending i to i • a.
Equations
- fin_equiv_multiples x = equiv.of_bijective (λ (n : fin (add_order_of x)), ⟨↑n • x, _⟩) _
The equivalence between fin (order_of x) and submonoid.powers x, sending i to x ^ i."
Equations
- fin_equiv_powers x = equiv.of_bijective (λ (n : fin (order_of x)), ⟨x ^ ↑n, _⟩) _
@[simp]
theorem
fin_equiv_multiples_apply
{G : Type u}
[fintype G]
[add_left_cancel_monoid G]
{x : G}
{n : fin (add_order_of x)} :
@[simp]
theorem
fin_equiv_powers_apply
{G : Type u}
[fintype G]
[left_cancel_monoid G]
{x : G}
{n : fin (order_of x)} :
@[simp]
theorem
fin_equiv_multiples_symm_apply
{G : Type u}
[fintype G]
[add_left_cancel_monoid G]
(x : G)
(n : ℕ)
{hn : ∃ (m : ℕ), m • x = n • x} :
⇑((fin_equiv_multiples x).symm) ⟨n • x, hn⟩ = ⟨n % add_order_of x, _⟩
noncomputable
def
powers_equiv_powers
{G : Type u}
{x y : G}
[fintype G]
[left_cancel_monoid G]
(h : order_of x = order_of y) :
↥↑(submonoid.powers x) ≃ ↥↑(submonoid.powers y)
The equivalence between submonoid.powers of two elements x, y of the same order, mapping
x ^ i to y ^ i.
Equations
- powers_equiv_powers h = (fin_equiv_powers x).symm.trans ((fin.cast h).to_equiv.trans (fin_equiv_powers y))
noncomputable
def
multiples_equiv_multiples
{G : Type u}
{x y : G}
[fintype G]
[add_left_cancel_monoid G]
(h : add_order_of x = add_order_of y) :
The equivalence between submonoid.multiples of two elements a, b of the same additive order,
mapping i • a to i • b.
Equations
- multiples_equiv_multiples h = (fin_equiv_multiples x).symm.trans ((fin.cast h).to_equiv.trans (fin_equiv_multiples y))
@[simp]
theorem
powers_equiv_powers_apply
{G : Type u}
{x y : G}
[fintype G]
[left_cancel_monoid G]
(h : order_of x = order_of y)
(n : ℕ) :
@[simp]
theorem
multiples_equiv_multiples_apply
{G : Type u}
{x y : G}
[fintype G]
[add_left_cancel_monoid G]
(h : add_order_of x = add_order_of y)
(n : ℕ) :
theorem
add_order_of_eq_card_multiples
{G : Type u}
{x : G}
[fintype G]
[add_left_cancel_monoid G]
[decidable_eq G] :
theorem
order_eq_card_powers
{G : Type u}
{x : G}
[fintype G]
[left_cancel_monoid G]
[decidable_eq G] :
theorem
mem_powers_iff_mem_zpowers
{G : Type u}
{x y : G}
[fintype G]
[group G] :
y ∈ submonoid.powers x ↔ y ∈ subgroup.zpowers x
theorem
powers_eq_zpowers
{G : Type u}
[fintype G]
[group G]
(x : G) :
↑(submonoid.powers x) = ↑(subgroup.zpowers x)
theorem
mem_zmultiples_iff_mem_range_add_order_of
{G : Type u}
{x y : G}
[fintype G]
[add_group G]
[decidable_eq G] :
y ∈ add_subgroup.zmultiples x ↔ y ∈ finset.image (λ (ᾰ : ℕ), ᾰ • x) (finset.range (add_order_of x))
theorem
mem_zpowers_iff_mem_range_order_of
{G : Type u}
{x y : G}
[fintype G]
[group G]
[decidable_eq G] :
y ∈ subgroup.zpowers x ↔ y ∈ finset.image (pow x) (finset.range (order_of x))
@[protected, instance]
noncomputable
def
decidable_zmultiples
{G : Type u}
{x : G}
[fintype G]
[add_group G]
[decidable_eq G] :
decidable_pred (λ (_x : G), _x ∈ add_subgroup.zmultiples x)
Equations
- decidable_zmultiples = decidable_zmultiples._proof_1.mpr (decidable_zmultiples._proof_2.mpr decidable_multiples)
@[protected, instance]
noncomputable
def
decidable_zpowers
{G : Type u}
{x : G}
[fintype G]
[group G]
[decidable_eq G] :
decidable_pred (λ (_x : G), _x ∈ subgroup.zpowers x)
Equations
- decidable_zpowers = decidable_zpowers._proof_1.mpr (decidable_zpowers._proof_2.mpr decidable_powers)
The equivalence between fin (order_of x) and subgroup.zpowers x, sending i to x ^ i.
Equations
- fin_equiv_zpowers x = (fin_equiv_powers x).trans (equiv.set.of_eq _)
noncomputable
def
fin_equiv_zmultiples
{G : Type u}
[fintype G]
[add_group G]
(x : G) :
fin (add_order_of x) ≃ ↥↑(add_subgroup.zmultiples x)
The equivalence between fin (add_order_of a) and subgroup.zmultiples a, sending i
to i • a.
Equations
@[simp]
theorem
fin_equiv_zmultiples_apply
{G : Type u}
{x : G}
[fintype G]
[add_group G]
{n : fin (add_order_of x)} :
noncomputable
def
zmultiples_equiv_zmultiples
{G : Type u}
{x y : G}
[fintype G]
[add_group G]
(h : add_order_of x = add_order_of y) :
The equivalence between subgroup.zmultiples of two elements a, b of the same additive order,
mapping i • a to i • b.
Equations
noncomputable
def
zpowers_equiv_zpowers
{G : Type u}
{x y : G}
[fintype G]
[group G]
(h : order_of x = order_of y) :
↥↑(subgroup.zpowers x) ≃ ↥↑(subgroup.zpowers y)
The equivalence between subgroup.zpowers of two elements x, y of the same order, mapping
x ^ i to y ^ i.
Equations
- zpowers_equiv_zpowers h = (fin_equiv_zpowers x).symm.trans ((fin.cast h).to_equiv.trans (fin_equiv_zpowers y))
@[simp]
theorem
zmultiples_equiv_zmultiples_apply
{G : Type u}
{x y : G}
[fintype G]
[add_group G]
(h : add_order_of x = add_order_of y)
(n : ℕ) :
theorem
add_order_eq_card_zmultiples
{G : Type u}
{x : G}
[fintype G]
[add_group G]
[decidable_eq G] :
theorem
order_of_dvd_card_univ
{G : Type u}
{x : G}
[fintype G]
[group G] :
order_of x ∣ fintype.card G
@[simp]
@[simp]
theorem
nsmul_eq_mod_card
{G : Type u}
{x : G}
[fintype G]
[add_group G]
(n : ℕ) :
n • x = (n % fintype.card G) • x
theorem
pow_eq_mod_card
{G : Type u}
{x : G}
[fintype G]
[group G]
(n : ℕ) :
x ^ n = x ^ (n % fintype.card G)
@[simp]
theorem
pow_coprime_symm_apply
{G : Type u}
{n : ℕ}
[fintype G]
[group G]
(h : (fintype.card G).coprime n)
(g : G) :
⇑((pow_coprime h).symm) g = g ^ (fintype.card G).gcd_b n
If gcd(|G|,n)=1 then the nth power map is a bijection
@[simp]
theorem
pow_coprime_apply
{G : Type u}
{n : ℕ}
[fintype G]
[group G]
(h : (fintype.card G).coprime n)
(g : G) :
⇑(pow_coprime h) g = g ^ n
@[simp]
theorem
nsmul_coprime_symm_apply
{G : Type u}
{n : ℕ}
[fintype G]
[add_group G]
(h : (fintype.card G).coprime n)
(g : G) :
⇑((nsmul_coprime h).symm) g = (fintype.card G).gcd_b n • g
@[simp]
theorem
nsmul_coprime_apply
{G : Type u}
{n : ℕ}
[fintype G]
[add_group G]
(h : (fintype.card G).coprime n)
(g : G) :
⇑(nsmul_coprime h) g = n • g
def
nsmul_coprime
{G : Type u}
{n : ℕ}
[fintype G]
[add_group G]
(h : (fintype.card G).coprime n) :
G ≃ G
If gcd(|G|,n)=1 then the smul by n is a bijection
@[simp]
theorem
pow_coprime_zero
{G : Type u}
{n : ℕ}
[fintype G]
[add_group G]
(h : (fintype.card G).coprime n) :
⇑(nsmul_coprime h) 0 = 0
@[simp]
theorem
pow_coprime_one
{G : Type u}
{n : ℕ}
[fintype G]
[group G]
(h : (fintype.card G).coprime n) :
⇑(pow_coprime h) 1 = 1
@[simp]
theorem
pow_coprime_inv
{G : Type u}
{n : ℕ}
[fintype G]
[group G]
(h : (fintype.card G).coprime n)
{g : G} :
⇑(pow_coprime h) g⁻¹ = (⇑(pow_coprime h) g)⁻¹
@[simp]
theorem
pow_coprime_neg
{G : Type u}
{n : ℕ}
[fintype G]
[add_group G]
(h : (fintype.card G).coprime n)
{g : G} :
⇑(nsmul_coprime h) (-g) = -⇑(nsmul_coprime h) g
theorem
add_inf_eq_bot_of_coprime
{G : Type u_1}
[add_group G]
{H K : add_subgroup G}
[fintype ↥H]
[fintype ↥K]
(h : (fintype.card ↥H).coprime (fintype.card ↥K)) :
theorem
image_range_add_order_of
{G : Type u}
{x : G}
[fintype G]
[add_group G]
[decidable_eq G] :
finset.image (λ (i : ℕ), i • x) (finset.range (add_order_of x)) = ↑(add_subgroup.zmultiples x).to_finset
theorem
image_range_order_of
{G : Type u}
{x : G}
[fintype G]
[group G]
[decidable_eq G] :
finset.image (λ (i : ℕ), x ^ i) (finset.range (order_of x)) = ↑(subgroup.zpowers x).to_finset
TODO: Generalise to submonoid.powers.
def
add_submonoid_of_idempotent
{M : Type u_1}
[add_left_cancel_monoid M]
[fintype M]
(S : set M)
(hS1 : S.nonempty)
(hS2 : S + S = S) :
A nonempty idempotent subset of a finite cancellative add monoid is a submonoid
def
submonoid_of_idempotent
{M : Type u_1}
[left_cancel_monoid M]
[fintype M]
(S : set M)
(hS1 : S.nonempty)
(hS2 : S * S = S) :
A nonempty idempotent subset of a finite cancellative monoid is a submonoid
@[simp]
theorem
smul_card_add_subgroup_coe
{G : Type u_1}
[add_group G]
[fintype G]
(S : set G)
(hS : S.nonempty) :
↑(smul_card_add_subgroup S hS) = fintype.card G • S
If S is a nonempty subset of a finite add group G,
then |G| • S is a subgroup
Equations
- smul_card_add_subgroup S hS = have one_mem : 0 ∈ fintype.card G • S, from _, add_subgroup_of_idempotent (fintype.card G • S) _ _
@[simp]
theorem
pow_card_subgroup_coe
{G : Type u_1}
[group G]
[fintype G]
(S : set G)
(hS : S.nonempty) :
↑(pow_card_subgroup S hS) = S ^ fintype.card G
def
pow_card_subgroup
{G : Type u_1}
[group G]
[fintype G]
(S : set G)
(hS : S.nonempty) :
subgroup G
If S is a nonempty subset of a finite group G, then S ^ |G| is a subgroup
Equations
- pow_card_subgroup S hS = have one_mem : 1 ∈ S ^ fintype.card G, from _, subgroup_of_idempotent (S ^ fintype.card G) _ _