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group_theory.order_of_element

Order of an element #

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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 #

Tags #

order of an element

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theorem is_periodic_pt_add_iff_nsmul_eq_zero {G : Type u_1} [add_monoid G] {n : ℕ} (x : G) :
function.is_periodic_pt (has_add.add x) n 0 ↔ n • x = 0
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theorem is_periodic_pt_mul_iff_pow_eq_one {G : Type u_1} [monoid G] {n : ℕ} (x : G) :
function.is_periodic_pt (has_mul.mul x) n 1 ↔ x ^ n = 1
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def is_of_fin_add_order {A : Type u_3} [add_monoid A] (a : A) :
Prop

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.

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def is_of_fin_order {G : Type u_1} [monoid G] (x : G) :
Prop

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.

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theorem is_of_fin_add_order_of_mul_iff {G : Type u_1} [monoid G] {x : G} :
is_of_fin_add_order (⇑additive.of_mul x) ↔ is_of_fin_order x
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theorem is_of_fin_order_of_add_iff {A : Type u_3} [add_monoid A] {a : A} :
is_of_fin_order (⇑multiplicative.of_add a) ↔ is_of_fin_add_order a
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theorem is_of_fin_order_iff_pow_eq_one {G : Type u_1} [monoid G] (x : G) :
is_of_fin_order x ↔ ∃ (n : ℕ), 0 < n ∧ x ^ n = 1
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theorem is_of_fin_add_order_iff_nsmul_eq_zero {G : Type u_1} [add_monoid G] (x : G) :
is_of_fin_add_order x ↔ ∃ (n : ℕ), 0 < n ∧ n • x = 0
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theorem not_is_of_fin_order_of_injective_pow {G : Type u_1} [monoid G] {x : G} (h : function.injective (λ (n : ℕ), x ^ n)) :
¬is_of_fin_order x

See also injective_pow_iff_not_is_of_fin_order.

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theorem not_is_of_fin_add_order_of_injective_nsmul {G : Type u_1} [add_monoid G] {x : G} (h : function.injective (λ (n : ℕ), n • x)) :
¬is_of_fin_add_order x

See also injective_nsmul_iff_not_is_of_fin_add_order.

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theorem is_of_fin_add_order_iff_coe {G : Type u_1} [add_monoid G] (H : add_submonoid G) (x : ↥H) :
is_of_fin_add_order x ↔ is_of_fin_add_order ↑x

Elements of finite order are of finite order in submonoids.

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theorem is_of_fin_order_iff_coe {G : Type u_1} [monoid G] (H : submonoid G) (x : ↥H) :
is_of_fin_order x ↔ is_of_fin_order ↑x

Elements of finite order are of finite order in submonoids.

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theorem add_monoid_hom.is_of_fin_order {G : Type u_1} {H : Type u_2} [add_monoid G] [add_monoid H] (f : G →+ H) {x : G} (h : is_of_fin_add_order x) :
is_of_fin_add_order (⇑f x)

The image of an element of finite additive order has finite additive order.

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theorem monoid_hom.is_of_fin_order {G : Type u_1} {H : Type u_2} [monoid G] [monoid H] (f : G →* H) {x : G} (h : is_of_fin_order x) :
is_of_fin_order (⇑f x)

The image of an element of finite order has finite order.

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theorem is_of_fin_order.apply {η : Type u_1} {Gs : η → Type u_2} [Π (i : η), monoid (Gs i)] {x : Π (i : η), Gs i} (h : is_of_fin_order x) (i : η) :
is_of_fin_order (x i)

If a direct product has finite order then so does each component.

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theorem is_of_fin_add_order.apply {η : Type u_1} {Gs : η → Type u_2} [Π (i : η), add_monoid (Gs i)] {x : Π (i : η), Gs i} (h : is_of_fin_add_order x) (i : η) :
is_of_fin_add_order (x i)

If a direct product has finite additive order then so does each component.

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theorem is_of_fin_order_zero {G : Type u_1} [add_monoid G] :
is_of_fin_add_order 0

0 is of finite order in any additive monoid.

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theorem is_of_fin_order_one {G : Type u_1} [monoid G] :
is_of_fin_order 1

1 is of finite order in any monoid.

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noncomputable def order_of {G : Type u_1} [monoid G] (x : G) :
ℕ

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.

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noncomputable def add_order_of {G : Type u_1} [add_monoid G] (x : G) :
ℕ

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.

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@[simp]
theorem add_order_of_of_mul_eq_order_of {G : Type u_1} [monoid G] (x : G) :
add_order_of (⇑additive.of_mul x) = order_of x
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@[simp]
theorem order_of_of_add_eq_add_order_of {A : Type u_3} [add_monoid A] (a : A) :
order_of (⇑multiplicative.of_add a) = add_order_of a
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theorem add_order_of_pos' {G : Type u_1} [add_monoid G] {x : G} (h : is_of_fin_add_order x) :
0 < add_order_of x
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theorem order_of_pos' {G : Type u_1} [monoid G] {x : G} (h : is_of_fin_order x) :
0 < order_of x
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theorem pow_order_of_eq_one {G : Type u_1} [monoid G] (x : G) :
x ^ order_of x = 1
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theorem add_order_of_nsmul_eq_zero {G : Type u_1} [add_monoid G] (x : G) :
add_order_of x • x = 0
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theorem order_of_eq_zero {G : Type u_1} [monoid G] {x : G} (h : ¬is_of_fin_order x) :
order_of x = 0
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theorem add_order_of_eq_zero {G : Type u_1} [add_monoid G] {x : G} (h : ¬is_of_fin_add_order x) :
add_order_of x = 0
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theorem add_order_of_eq_zero_iff {G : Type u_1} [add_monoid G] {x : G} :
add_order_of x = 0 ↔ ¬is_of_fin_add_order x
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theorem order_of_eq_zero_iff {G : Type u_1} [monoid G] {x : G} :
order_of x = 0 ↔ ¬is_of_fin_order x
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theorem order_of_eq_zero_iff' {G : Type u_1} [monoid G] {x : G} :
order_of x = 0 ↔ ∀ (n : ℕ), 0 < n → x ^ n ≠ 1
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theorem add_order_of_eq_zero_iff' {G : Type u_1} [add_monoid G] {x : G} :
add_order_of x = 0 ↔ ∀ (n : ℕ), 0 < n → n • x ≠ 0
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theorem add_order_of_eq_iff {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} (h : 0 < n) :
add_order_of x = n ↔ n • x = 0 ∧ ∀ (m : ℕ), m < n → 0 < m → m • x ≠ 0
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theorem order_of_eq_iff {G : Type u_1} [monoid G] {x : G} {n : ℕ} (h : 0 < n) :
order_of x = n ↔ x ^ n = 1 ∧ ∀ (m : ℕ), m < n → 0 < m → x ^ m ≠ 1
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theorem add_order_of_pos_iff {G : Type u_1} [add_monoid G] {x : G} :
0 < add_order_of x ↔ is_of_fin_add_order x

A group element has finite additive order iff its order is positive.

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theorem order_of_pos_iff {G : Type u_1} [monoid G] {x : G} :
0 < order_of x ↔ is_of_fin_order x

A group element has finite order iff its order is positive.

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theorem is_of_fin_add_order.mono {G : Type u_1} {β : Type u_5} [add_monoid G] {x : G} [add_monoid β] {y : β} (hx : is_of_fin_add_order x) (h : add_order_of y ∣ add_order_of x) :
is_of_fin_add_order y
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theorem is_of_fin_order.mono {G : Type u_1} {β : Type u_5} [monoid G] {x : G} [monoid β] {y : β} (hx : is_of_fin_order x) (h : order_of y ∣ order_of x) :
is_of_fin_order y
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theorem pow_ne_one_of_lt_order_of' {G : Type u_1} [monoid G] {x : G} {n : ℕ} (n0 : n ≠ 0) (h : n < order_of x) :
x ^ n ≠ 1
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theorem nsmul_ne_zero_of_lt_add_order_of' {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} (n0 : n ≠ 0) (h : n < add_order_of x) :
n • x ≠ 0
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theorem add_order_of_le_of_nsmul_eq_zero {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} (hn : 0 < n) (h : n • x = 0) :
add_order_of x ≤ n
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theorem order_of_le_of_pow_eq_one {G : Type u_1} [monoid G] {x : G} {n : ℕ} (hn : 0 < n) (h : x ^ n = 1) :
order_of x ≤ n
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@[simp]
theorem order_of_one {G : Type u_1} [monoid G] :
order_of 1 = 1
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@[simp]
theorem order_of_zero {G : Type u_1} [add_monoid G] :
add_order_of 0 = 1
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@[simp]
theorem order_of_eq_one_iff {G : Type u_1} [monoid G] {x : G} :
order_of x = 1 ↔ x = 1
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@[simp]
theorem add_monoid.order_of_eq_one_iff {G : Type u_1} [add_monoid G] {x : G} :
add_order_of x = 1 ↔ x = 0
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theorem pow_eq_mod_order_of {G : Type u_1} [monoid G] {x : G} {n : ℕ} :
x ^ n = x ^ (n % order_of x)
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theorem nsmul_eq_mod_add_order_of {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} :
n • x = (n % add_order_of x) • x
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theorem order_of_dvd_of_pow_eq_one {G : Type u_1} [monoid G] {x : G} {n : ℕ} (h : x ^ n = 1) :
order_of x ∣ n
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theorem add_order_of_dvd_of_nsmul_eq_zero {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} (h : n • x = 0) :
add_order_of x ∣ n
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theorem add_order_of_dvd_iff_nsmul_eq_zero {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} :
add_order_of x ∣ n ↔ n • x = 0
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theorem order_of_dvd_iff_pow_eq_one {G : Type u_1} [monoid G] {x : G} {n : ℕ} :
order_of x ∣ n ↔ x ^ n = 1
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theorem add_order_of_smul_dvd {G : Type u_1} [add_monoid G] {x : G} (n : ℕ) :
add_order_of (n • x) ∣ add_order_of x
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theorem order_of_pow_dvd {G : Type u_1} [monoid G] {x : G} (n : ℕ) :
order_of (x ^ n) ∣ order_of x
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theorem add_order_of_map_dvd {G : Type u_1} [add_monoid G] {H : Type u_2} [add_monoid H] (ψ : G →+ H) (x : G) :
add_order_of (⇑ψ x) ∣ add_order_of x
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theorem order_of_map_dvd {G : Type u_1} [monoid G] {H : Type u_2} [monoid H] (ψ : G →* H) (x : G) :
order_of (⇑ψ x) ∣ order_of x
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theorem exists_pow_eq_self_of_coprime {G : Type u_1} [monoid G] {x : G} {n : ℕ} (h : n.coprime (order_of x)) :
∃ (m : ℕ), (x ^ n) ^ m = x
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theorem exists_nsmul_eq_self_of_coprime {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} (h : n.coprime (add_order_of x)) :
∃ (m : ℕ), m • n • x = x
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theorem add_order_of_eq_of_nsmul_and_div_prime_nsmul {G : Type u_1} [add_monoid G] {x : G} {n : ℕ} (hn : 0 < n) (hx : n • x = 0) (hd : ∀ (p : ℕ), nat.prime p → p ∣ n → (n / p) • x ≠ 0) :
add_order_of x = n

If n * x = 0, but n/p * x ≠ 0 for all prime factors p of n, then x has order n in G.

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theorem order_of_eq_of_pow_and_pow_div_prime {G : Type u_1} [monoid G] {x : G} {n : ℕ} (hn : 0 < n) (hx : x ^ n = 1) (hd : ∀ (p : ℕ), nat.prime p → p ∣ n → x ^ (n / p) ≠ 1) :
order_of x = n

If x^n = 1, but x^(n/p) ≠ 1 for all prime factors p of n, then x has order n in G.

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theorem order_of_eq_order_of_iff {G : Type u_1} [monoid G] {x : G} {H : Type u_2} [monoid H] {y : H} :
order_of x = order_of y ↔ ∀ (n : ℕ), x ^ n = 1 ↔ y ^ n = 1
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theorem add_order_of_eq_add_order_of_iff {G : Type u_1} [add_monoid G] {x : G} {H : Type u_2} [add_monoid H] {y : H} :
add_order_of x = add_order_of y ↔ ∀ (n : ℕ), n • x = 0 ↔ n • y = 0
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theorem order_of_injective {G : Type u_1} [monoid G] {H : Type u_2} [monoid H] (f : G →* H) (hf : function.injective ⇑f) (x : G) :
order_of (⇑f x) = order_of x
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theorem add_order_of_injective {G : Type u_1} [add_monoid G] {H : Type u_2} [add_monoid H] (f : G →+ H) (hf : function.injective ⇑f) (x : G) :
add_order_of (⇑f x) = add_order_of x
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@[simp, norm_cast]
theorem order_of_submonoid {G : Type u_1} [monoid G] {H : submonoid G} (y : ↥H) :
order_of ↑y = order_of y
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@[simp, norm_cast]
theorem order_of_add_submonoid {G : Type u_1} [add_monoid G] {H : add_submonoid G} (y : ↥H) :
add_order_of ↑y = add_order_of y
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theorem order_of_units {G : Type u_1} [monoid G] {y : Gˣ} :
order_of ↑y = order_of y
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theorem order_of_add_units {G : Type u_1} [add_monoid G] {y : add_units G} :
add_order_of ↑y = add_order_of y
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theorem add_order_of_nsmul' {G : Type u_1} [add_monoid G] (x : G) {n : ℕ} (h : n ≠ 0) :
add_order_of (n • x) = add_order_of x / (add_order_of x).gcd n
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theorem order_of_pow' {G : Type u_1} [monoid G] (x : G) {n : ℕ} (h : n ≠ 0) :
order_of (x ^ n) = order_of x / (order_of x).gcd n
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theorem add_order_of_nsmul'' {G : Type u_1} [add_monoid G] (x : G) (n : ℕ) (h : is_of_fin_add_order x) :
add_order_of (n • x) = add_order_of x / (add_order_of x).gcd n
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theorem order_of_pow'' {G : Type u_1} [monoid G] (x : G) (n : ℕ) (h : is_of_fin_order x) :
order_of (x ^ n) = order_of x / (order_of x).gcd n
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theorem add_order_of_nsmul_coprime {G : Type u_1} [add_monoid G] {y : G} {m : ℕ} (h : (add_order_of y).coprime m) :
add_order_of (m • y) = add_order_of y
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theorem order_of_pow_coprime {G : Type u_1} [monoid G] {y : G} {m : ℕ} (h : (order_of y).coprime m) :
order_of (y ^ m) = order_of y
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theorem commute.order_of_mul_dvd_lcm {G : Type u_1} [monoid G] {x y : G} (h : commute x y) :
order_of (x * y) ∣ (order_of x).lcm (order_of y)
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theorem add_commute.order_of_add_dvd_lcm {G : Type u_1} [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)
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theorem commute.order_of_dvd_lcm_mul {G : Type u_1} [monoid G] {x y : G} (h : commute x y) :
order_of y ∣ (order_of x).lcm (order_of (x * y))
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theorem add_commute.order_of_dvd_lcm_add {G : Type u_1} [add_monoid G] {x y : G} (h : add_commute x y) :
add_order_of y ∣ (add_order_of x).lcm (add_order_of (x + y))
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theorem commute.order_of_mul_dvd_mul_order_of {G : Type u_1} [monoid G] {x y : G} (h : commute x y) :
order_of (x * y) ∣ order_of x * order_of y
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theorem add_commute.add_order_of_add_dvd_mul_add_order_of {G : Type u_1} [add_monoid G] {x y : G} (h : add_commute x y) :
add_order_of (x + y) ∣ add_order_of x * add_order_of y
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theorem add_commute.add_order_of_add_eq_mul_add_order_of_of_coprime {G : Type u_1} [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
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theorem commute.order_of_mul_eq_mul_order_of_of_coprime {G : Type u_1} [monoid G] {x y : G} (h : commute x y) (hco : (order_of x).coprime (order_of y)) :
order_of (x * y) = order_of x * order_of y
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theorem add_commute.is_of_fin_order_add {G : Type u_1} [add_monoid G] {x y : G} (h : add_commute x y) (hx : is_of_fin_add_order x) (hy : is_of_fin_add_order y) :
is_of_fin_add_order (x + y)

Commuting elements of finite additive order are closed under addition.

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theorem commute.is_of_fin_order_mul {G : Type u_1} [monoid G] {x y : G} (h : commute x y) (hx : is_of_fin_order x) (hy : is_of_fin_order y) :
is_of_fin_order (x * y)

Commuting elements of finite order are closed under multiplication.

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theorem add_commute.add_order_of_add_eq_right_of_forall_prime_mul_dvd {G : Type u_1} [add_monoid G] {x y : G} (h : add_commute x y) (hy : is_of_fin_add_order y) (hdvd : ∀ (p : ℕ), nat.prime p → p ∣ add_order_of x → p * add_order_of x ∣ add_order_of y) :
add_order_of (x + y) = add_order_of y

If each prime factor of add_order_of x has higher multiplicity in add_order_of y, and x commutes with y, then x + y has the same order as y.

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theorem commute.order_of_mul_eq_right_of_forall_prime_mul_dvd {G : Type u_1} [monoid G] {x y : G} (h : commute x y) (hy : is_of_fin_order y) (hdvd : ∀ (p : ℕ), nat.prime p → p ∣ order_of x → p * order_of x ∣ order_of y) :
order_of (x * y) = order_of y

If each prime factor of order_of x has higher multiplicity in order_of y, and x commutes with y, then x * y has the same order as y.

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theorem add_order_of_eq_prime {G : Type u_1} [add_monoid G] {x : G} {p : ℕ} [hp : fact (nat.prime p)] (hg : p • x = 0) (hg1 : x ≠ 0) :
add_order_of x = p
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theorem order_of_eq_prime {G : Type u_1} [monoid G] {x : G} {p : ℕ} [hp : fact (nat.prime p)] (hg : x ^ p = 1) (hg1 : x ≠ 1) :
order_of x = p
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theorem add_order_of_eq_prime_pow {G : Type u_1} [add_monoid G] {x : G} {n p : ℕ} [hp : fact (nat.prime p)] (hnot : ¬p ^ n • x = 0) (hfin : p ^ (n + 1) • x = 0) :
add_order_of x = p ^ (n + 1)
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theorem order_of_eq_prime_pow {G : Type u_1} [monoid G] {x : G} {n p : ℕ} [hp : fact (nat.prime p)] (hnot : ¬x ^ p ^ n = 1) (hfin : x ^ p ^ (n + 1) = 1) :
order_of x = p ^ (n + 1)
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theorem exists_add_order_of_eq_prime_pow_iff {G : Type u_1} [add_monoid G] {x : G} {p : ℕ} [hp : fact (nat.prime p)] :
(∃ (k : ℕ), add_order_of x = p ^ k) ↔ ∃ (m : ℕ), p ^ m • x = 0
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theorem exists_order_of_eq_prime_pow_iff {G : Type u_1} [monoid G] {x : G} {p : ℕ} [hp : fact (nat.prime p)] :
(∃ (k : ℕ), order_of x = p ^ k) ↔ ∃ (m : ℕ), x ^ p ^ m = 1
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theorem nsmul_injective_of_lt_add_order_of {G : Type u_1} [add_left_cancel_monoid G] (x : G) {m n : ℕ} (hn : n < add_order_of x) (hm : m < add_order_of x) (eq : n • x = m • x) :
n = m
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theorem pow_injective_of_lt_order_of {G : Type u_1} [left_cancel_monoid G] (x : G) {m n : ℕ} (hn : n < order_of x) (hm : m < order_of x) (eq : x ^ n = x ^ m) :
n = m
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theorem mem_powers_iff_mem_range_order_of' {G : Type u_1} [left_cancel_monoid G] (x y : G) [decidable_eq G] (hx : 0 < order_of x) :
y ∈ submonoid.powers x ↔ y ∈ finset.image (has_pow.pow x) (finset.range (order_of x))
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theorem mem_multiples_iff_mem_range_add_order_of' {G : Type u_1} [add_left_cancel_monoid G] (x y : 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))
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theorem pow_eq_one_iff_modeq {G : Type u_1} [left_cancel_monoid G] (x : G) {n : ℕ} :
x ^ n = 1 ↔ n ≡ 0 [MOD order_of x]
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theorem nsmul_eq_zero_iff_modeq {G : Type u_1} [add_left_cancel_monoid G] (x : G) {n : ℕ} :
n • x = 0 ↔ n ≡ 0 [MOD add_order_of x]
source
theorem pow_eq_pow_iff_modeq {G : Type u_1} [left_cancel_monoid G] (x : G) {m n : ℕ} :
x ^ n = x ^ m ↔ n ≡ m [MOD order_of x]
source
theorem nsmul_eq_nsmul_iff_modeq {G : Type u_1} [add_left_cancel_monoid G] (x : G) {m n : ℕ} :
n • x = m • x ↔ n ≡ m [MOD add_order_of x]
source
@[simp]
theorem injective_pow_iff_not_is_of_fin_order {G : Type u_1} [left_cancel_monoid G] {x : G} :
function.injective (λ (n : ℕ), x ^ n) ↔ ¬is_of_fin_order x
source
@[simp]
theorem injective_nsmul_iff_not_is_of_fin_add_order {G : Type u_1} [add_left_cancel_monoid G] {x : G} :
function.injective (λ (n : ℕ), n • x) ↔ ¬is_of_fin_add_order x
source
theorem infinite_not_is_of_fin_add_order {G : Type u_1} [add_left_cancel_monoid G] {x : G} (h : ¬is_of_fin_add_order x) :
{y : G | ¬is_of_fin_add_order y}.infinite
source
theorem infinite_not_is_of_fin_order {G : Type u_1} [left_cancel_monoid G] {x : G} (h : ¬is_of_fin_order x) :
{y : G | ¬is_of_fin_order y}.infinite
source
theorem is_of_fin_order.inv {G : Type u_1} [group G] {x : G} (hx : is_of_fin_order x) :
is_of_fin_order x⁻¹

Inverses of elements of finite order have finite order.

source
theorem is_of_fin_add_order.neg {G : Type u_1} [add_group G] {x : G} (hx : is_of_fin_add_order x) :
is_of_fin_add_order (-x)

Inverses of elements of finite additive order have finite additive order.

source
@[simp]
theorem is_of_fin_order_inv_iff {G : Type u_1} [group G] {x : G} :
is_of_fin_order x⁻¹ ↔ is_of_fin_order x

Inverses of elements of finite order have finite order.

source
@[simp]
theorem is_of_fin_order_neg_iff {G : Type u_1} [add_group G] {x : G} :
is_of_fin_add_order (-x) ↔ is_of_fin_add_order x

Inverses of elements of finite additive order have finite additive order.

source
theorem order_of_dvd_iff_zpow_eq_one {G : Type u_1} [group G] {x : G} {i : ℤ} :
↑(order_of x) ∣ i ↔ x ^ i = 1
source
theorem add_order_of_dvd_iff_zsmul_eq_zero {G : Type u_1} [add_group G] {x : G} {i : ℤ} :
↑(add_order_of x) ∣ i ↔ i • x = 0
source
@[simp]
theorem order_of_neg {G : Type u_1} [add_group G] (x : G) :
add_order_of (-x) = add_order_of x
source
@[simp]
theorem order_of_inv {G : Type u_1} [group G] (x : G) :
order_of x⁻¹ = order_of x
source
@[simp, norm_cast]
theorem order_of_add_subgroup {G : Type u_1} [add_group G] {H : add_subgroup G} (y : ↥H) :
add_order_of ↑y = add_order_of y
source
@[simp, norm_cast]
theorem order_of_subgroup {G : Type u_1} [group G] {H : subgroup G} (y : ↥H) :
order_of ↑y = order_of y
source
theorem zsmul_eq_mod_add_order_of {G : Type u_1} [add_group G] {x : G} {i : ℤ} :
i • x = (i % ↑(add_order_of x)) • x
source
theorem zpow_eq_mod_order_of {G : Type u_1} [group G] {x : G} {i : ℤ} :
x ^ i = x ^ (i % ↑(order_of x))
source
theorem pow_inj_iff_of_order_of_eq_zero {G : Type u_1} [group G] {x : G} (h : order_of x = 0) {n m : ℕ} :
x ^ n = x ^ m ↔ n = m
source
theorem nsmul_inj_iff_of_add_order_of_eq_zero {G : Type u_1} [add_group G] {x : G} (h : add_order_of x = 0) {n m : ℕ} :
n • x = m • x ↔ n = m
source
theorem nsmul_inj_mod {G : Type u_1} [add_group G] {x : G} {n m : ℕ} :
n • x = m • x ↔ n % add_order_of x = m % add_order_of x
source
theorem pow_inj_mod {G : Type u_1} [group G] {x : G} {n m : ℕ} :
x ^ n = x ^ m ↔ n % order_of x = m % order_of x
source
@[simp]
theorem zpow_pow_order_of {G : Type u_1} [group G] {x : G} {i : ℤ} :
(x ^ i) ^ order_of x = 1
source
@[simp]
theorem zsmul_smul_order_of {G : Type u_1} [add_group G] {x : G} {i : ℤ} :
add_order_of x • i • x = 0
source
theorem is_of_fin_add_order.zsmul {G : Type u_1} [add_group G] {x : G} (h : is_of_fin_add_order x) {i : ℤ} :
is_of_fin_add_order (i • x)
source
theorem is_of_fin_order.zpow {G : Type u_1} [group G] {x : G} (h : is_of_fin_order x) {i : ℤ} :
is_of_fin_order (x ^ i)
source
theorem is_of_fin_order.of_mem_zpowers {G : Type u_1} [group G] {x y : G} (h : is_of_fin_order x) (h' : y ∈ subgroup.zpowers x) :
is_of_fin_order y
source
theorem is_of_fin_add_order.of_mem_zmultiples {G : Type u_1} [add_group G] {x y : G} (h : is_of_fin_add_order x) (h' : y ∈ add_subgroup.zmultiples x) :
is_of_fin_add_order y
source
theorem add_order_of_dvd_of_mem_zmultiples {G : Type u_1} [add_group G] {x y : G} (h : y ∈ add_subgroup.zmultiples x) :
add_order_of y ∣ add_order_of x
source
theorem order_of_dvd_of_mem_zpowers {G : Type u_1} [group G] {x y : G} (h : y ∈ subgroup.zpowers x) :
order_of y ∣ order_of x
source
theorem smul_eq_self_of_mem_zpowers {G : Type u_1} [group G] {x y : G} {α : Type u_2} [mul_action G α] (hx : x ∈ subgroup.zpowers y) {a : α} (hs : y • a = a) :
x • a = a
source
theorem vadd_eq_self_of_mem_zmultiples {α : Type u_1} {G : Type u_2} [add_group G] [add_action G α] {x y : G} (hx : x ∈ add_subgroup.zmultiples y) {a : α} (hs : y +ᵥ a = a) :
x +ᵥ a = a
source
theorem is_of_fin_order.mul {G : Type u_1} [comm_monoid G] {x y : G} (hx : is_of_fin_order x) (hy : is_of_fin_order y) :
is_of_fin_order (x * y)

Elements of finite order are closed under multiplication.

source
theorem is_of_fin_add_order.add {G : Type u_1} [add_comm_monoid G] {x y : G} (hx : is_of_fin_add_order x) (hy : is_of_fin_add_order y) :
is_of_fin_add_order (x + y)

Elements of finite additive order are closed under addition.

source
theorem sum_card_order_of_eq_card_pow_eq_one {G : Type u_1} [monoid G] {n : ℕ} [fintype G] [decidable_eq G] (hn : n ≠ 0) :
(finset.filter (λ (_x : ℕ), _x ∣ n) (finset.range n.succ)).sum (λ (m : ℕ), (finset.filter (λ (x : G), order_of x = m) finset.univ).card) = (finset.filter (λ (x : G), x ^ n = 1) finset.univ).card
source
theorem sum_card_add_order_of_eq_card_nsmul_eq_zero {G : Type u_1} [add_monoid G] {n : ℕ} [fintype G] [decidable_eq G] (hn : n ≠ 0) :
(finset.filter (λ (_x : ℕ), _x ∣ n) (finset.range n.succ)).sum (λ (m : ℕ), (finset.filter (λ (x : G), add_order_of x = m) finset.univ).card) = (finset.filter (λ (x : G), n • x = 0) finset.univ).card
source
theorem exists_pow_eq_one {G : Type u_1} [left_cancel_monoid G] [finite G] (x : G) :
is_of_fin_order x
source
theorem exists_nsmul_eq_zero {G : Type u_1} [add_left_cancel_monoid G] [finite G] (x : G) :
is_of_fin_add_order x
source
theorem order_of_le_card_univ {G : Type u_1} [left_cancel_monoid G] {x : G} [fintype G] :
order_of x ≤ fintype.card G
source
theorem add_order_of_le_card_univ {G : Type u_1} [add_left_cancel_monoid G] {x : G} [fintype G] :
add_order_of x ≤ fintype.card G
source
theorem order_of_pos {G : Type u_1} [left_cancel_monoid G] [finite G] (x : G) :
0 < order_of x

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.

source
theorem add_order_of_pos {G : Type u_1} [add_left_cancel_monoid G] [finite 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.

source
theorem add_order_of_nsmul {G : Type u_1} [add_left_cancel_monoid G] {n : ℕ} [finite 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.

source
theorem order_of_pow {G : Type u_1} [left_cancel_monoid G] {n : ℕ} [finite G] (x : G) :
order_of (x ^ n) = order_of x / (order_of x).gcd n

This is the same as order_of_pow' and order_of_pow'' but with one assumption less which is automatic in the case of a finite cancellative monoid.

source
theorem mem_multiples_iff_mem_range_add_order_of {G : Type u_1} [add_left_cancel_monoid G] {x y : G} [finite G] [decidable_eq G] :
y ∈ add_submonoid.multiples x ↔ y ∈ finset.image (λ (ᾰ : ℕ), ᾰ • x) (finset.range (add_order_of x))
source
theorem mem_powers_iff_mem_range_order_of {G : Type u_1} [left_cancel_monoid G] {x y : G} [finite G] [decidable_eq G] :
y ∈ submonoid.powers x ↔ y ∈ finset.image (has_pow.pow x) (finset.range (order_of x))
source
@[protected, instance]
noncomputable def decidable_multiples {G : Type u_1} [add_left_cancel_monoid G] {x : G} :
decidable_pred (λ (_x : G), _x ∈ add_submonoid.multiples x)
Equations
source
@[protected, instance]
noncomputable def decidable_powers {G : Type u_1} [left_cancel_monoid G] {x : G} :
decidable_pred (λ (_x : G), _x ∈ submonoid.powers x)
Equations
source
noncomputable def fin_equiv_multiples {G : Type u_1} [add_left_cancel_monoid G] [finite 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
source
noncomputable def fin_equiv_powers {G : Type u_1} [left_cancel_monoid G] [finite G] (x : G) :
fin (order_of x) ≃ ↥↑(submonoid.powers x)

The equivalence between fin (order_of x) and submonoid.powers x, sending i to x ^ i."

Equations
source
@[simp]
theorem fin_equiv_multiples_apply {G : Type u_1} [add_left_cancel_monoid G] [finite G] {x : G} {n : fin (add_order_of x)} :
⇑(fin_equiv_multiples x) n = ⟨↑n • x, _⟩
source
@[simp]
theorem fin_equiv_powers_apply {G : Type u_1} [left_cancel_monoid G] [finite G] {x : G} {n : fin (order_of x)} :
⇑(fin_equiv_powers x) n = ⟨x ^ ↑n, _⟩
source
@[simp]
theorem fin_equiv_multiples_symm_apply {G : Type u_1} [add_left_cancel_monoid G] [finite 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, _⟩
source
@[simp]
theorem fin_equiv_powers_symm_apply {G : Type u_1} [left_cancel_monoid G] [finite G] (x : G) (n : ℕ) {hn : ∃ (m : ℕ), x ^ m = x ^ n} :
⇑((fin_equiv_powers x).symm) ⟨x ^ n, hn⟩ = ⟨n % order_of x, _⟩
source
noncomputable def powers_equiv_powers {G : Type u_1} [left_cancel_monoid G] {x y : G} [finite 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
source
noncomputable def multiples_equiv_multiples {G : Type u_1} [add_left_cancel_monoid G] {x y : G} [finite G] (h : add_order_of x = add_order_of y) :
↥↑(add_submonoid.multiples x) ≃ ↥↑(add_submonoid.multiples y)

The equivalence between submonoid.multiples of two elements a, b of the same additive order, mapping i • a to i • b.

Equations
source
@[simp]
theorem powers_equiv_powers_apply {G : Type u_1} [left_cancel_monoid G] {x y : G} [finite G] (h : order_of x = order_of y) (n : ℕ) :
⇑(powers_equiv_powers h) ⟨x ^ n, _⟩ = ⟨y ^ n, _⟩
source
@[simp]
theorem multiples_equiv_multiples_apply {G : Type u_1} [add_left_cancel_monoid G] {x y : G} [finite G] (h : add_order_of x = add_order_of y) (n : ℕ) :
⇑(multiples_equiv_multiples h) ⟨n • x, _⟩ = ⟨n • y, _⟩
source
theorem add_order_of_eq_card_multiples {G : Type u_1} [add_left_cancel_monoid G] {x : G} [fintype G] :
add_order_of x = fintype.card ↥↑(add_submonoid.multiples x)
source
theorem order_eq_card_powers {G : Type u_1} [left_cancel_monoid G] {x : G} [fintype G] :
order_of x = fintype.card ↥↑(submonoid.powers x)
source
theorem exists_zpow_eq_one {G : Type u_1} [group G] [finite G] (x : G) :
∃ (i : ℤ) (H : i ≠ 0), x ^ i = 1
source
theorem exists_zsmul_eq_zero {G : Type u_1} [add_group G] [finite G] (x : G) :
∃ (i : ℤ) (H : i ≠ 0), i • x = 0
source
theorem mem_multiples_iff_mem_zmultiples {G : Type u_1} [add_group G] {x y : G} [finite G] :
y ∈ add_submonoid.multiples x ↔ y ∈ add_subgroup.zmultiples x
source
theorem mem_powers_iff_mem_zpowers {G : Type u_1} [group G] {x y : G} [finite G] :
y ∈ submonoid.powers x ↔ y ∈ subgroup.zpowers x
source
theorem multiples_eq_zmultiples {G : Type u_1} [add_group G] [finite G] (x : G) :
↑(add_submonoid.multiples x) = ↑(add_subgroup.zmultiples x)
source
theorem powers_eq_zpowers {G : Type u_1} [group G] [finite G] (x : G) :
↑(submonoid.powers x) = ↑(subgroup.zpowers x)
source
theorem mem_zmultiples_iff_mem_range_add_order_of {G : Type u_1} [add_group G] {x y : G} [finite G] [decidable_eq G] :
y ∈ add_subgroup.zmultiples x ↔ y ∈ finset.image (λ (ᾰ : ℕ), ᾰ • x) (finset.range (add_order_of x))
source
theorem mem_zpowers_iff_mem_range_order_of {G : Type u_1} [group G] {x y : G} [finite G] [decidable_eq G] :
y ∈ subgroup.zpowers x ↔ y ∈ finset.image (has_pow.pow x) (finset.range (order_of x))
source
theorem zsmul_eq_zero_iff_modeq {G : Type u_1} [add_group G] {x : G} {n : ℤ} :
n • x = 0 ↔ n ≡ 0 [ZMOD ↑(add_order_of x)]
source
theorem zpow_eq_one_iff_modeq {G : Type u_1} [group G] {x : G} {n : ℤ} :
x ^ n = 1 ↔ n ≡ 0 [ZMOD ↑(order_of x)]
source
theorem zpow_eq_zpow_iff_modeq {G : Type u_1} [group G] {x : G} {m n : ℤ} :
x ^ m = x ^ n ↔ m ≡ n [ZMOD ↑(order_of x)]
source
theorem zsmul_eq_zsmul_iff_modeq {G : Type u_1} [add_group G] {x : G} {m n : ℤ} :
m • x = n • x ↔ m ≡ n [ZMOD ↑(add_order_of x)]
source
@[simp]
theorem injective_zsmul_iff_not_is_of_fin_order {G : Type u_1} [add_group G] {x : G} :
function.injective (λ (n : ℤ), n • x) ↔ ¬is_of_fin_add_order x
source
@[simp]
theorem injective_zpow_iff_not_is_of_fin_order {G : Type u_1} [group G] {x : G} :
function.injective (λ (n : ℤ), x ^ n) ↔ ¬is_of_fin_order x
source
@[protected, instance]
noncomputable def decidable_zmultiples {G : Type u_1} [add_group G] {x : G} :
decidable_pred (λ (_x : G), _x ∈ add_subgroup.zmultiples x)
Equations
source
@[protected, instance]
noncomputable def decidable_zpowers {G : Type u_1} [group G] {x : G} :
decidable_pred (λ (_x : G), _x ∈ subgroup.zpowers x)
Equations
source
noncomputable def fin_equiv_zpowers {G : Type u_1} [group G] [finite G] (x : G) :
fin (order_of x) ≃ ↥↑(subgroup.zpowers x)

The equivalence between fin (order_of x) and subgroup.zpowers x, sending i to x ^ i.

Equations
source
noncomputable def fin_equiv_zmultiples {G : Type u_1} [add_group G] [finite 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
source
@[simp]
theorem fin_equiv_zpowers_apply {G : Type u_1} [group G] {x : G} [finite G] {n : fin (order_of x)} :
⇑(fin_equiv_zpowers x) n = ⟨x ^ ↑n, _⟩
source
@[simp]
theorem fin_equiv_zmultiples_apply {G : Type u_1} [add_group G] {x : G} [finite G] {n : fin (add_order_of x)} :
⇑(fin_equiv_zmultiples x) n = ⟨↑n • x, _⟩
source
@[simp]
theorem fin_equiv_zmultiples_symm_apply {G : Type u_1} [add_group G] [finite G] (x : G) (n : ℕ) {hn : ∃ (m : ℤ), m • x = n • x} :
⇑((fin_equiv_zmultiples x).symm) ⟨n • x, hn⟩ = ⟨n % add_order_of x, _⟩
source
@[simp]
theorem fin_equiv_zpowers_symm_apply {G : Type u_1} [group G] [finite G] (x : G) (n : ℕ) {hn : ∃ (m : ℤ), x ^ m = x ^ n} :
⇑((fin_equiv_zpowers x).symm) ⟨x ^ n, hn⟩ = ⟨n % order_of x, _⟩
source
noncomputable def zmultiples_equiv_zmultiples {G : Type u_1} [add_group G] {x y : G} [finite G] (h : add_order_of x = add_order_of y) :
↥↑(add_subgroup.zmultiples x) ≃ ↥↑(add_subgroup.zmultiples y)

The equivalence between subgroup.zmultiples of two elements a, b of the same additive order, mapping i • a to i • b.

Equations
source
noncomputable def zpowers_equiv_zpowers {G : Type u_1} [group G] {x y : G} [finite 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
source
@[simp]
theorem zmultiples_equiv_zmultiples_apply {G : Type u_1} [add_group G] {x y : G} [finite G] (h : add_order_of x = add_order_of y) (n : ℕ) :
⇑(zmultiples_equiv_zmultiples h) ⟨n • x, _⟩ = ⟨n • y, _⟩
source
@[simp]
theorem zpowers_equiv_zpowers_apply {G : Type u_1} [group G] {x y : G} [finite G] (h : order_of x = order_of y) (n : ℕ) :
⇑(zpowers_equiv_zpowers h) ⟨x ^ n, _⟩ = ⟨y ^ n, _⟩
source
theorem add_order_eq_card_zmultiples {G : Type u_1} [add_group G] {x : G} [fintype G] :
add_order_of x = fintype.card ↥(add_subgroup.zmultiples x)

See also nat.card_zmultiples.

source
theorem order_eq_card_zpowers {G : Type u_1} [group G] {x : G} [fintype G] :
order_of x = fintype.card ↥(subgroup.zpowers x)

See also nat.card_zpowers'.

source
theorem order_of_dvd_card_univ {G : Type u_1} [group G] {x : G} [fintype G] :
order_of x ∣ fintype.card G
source
theorem add_order_of_dvd_card_univ {G : Type u_1} [add_group G] {x : G} [fintype G] :
add_order_of x ∣ fintype.card G
source
theorem add_order_of_dvd_nat_card {G : Type u_1} [add_group G] {x : G} :
add_order_of x ∣ nat.card G
source
theorem order_of_dvd_nat_card {G : Type u_1} [group G] {x : G} :
order_of x ∣ nat.card G
source
@[simp]
theorem pow_card_eq_one' {G : Type u_1} [group G] {x : G} :
x ^ nat.card G = 1
source
@[simp]
theorem card_nsmul_eq_zero' {G : Type u_1} [add_group G] {x : G} :
nat.card G • x = 0
source
@[simp]
theorem pow_card_eq_one {G : Type u_1} [group G] {x : G} [fintype G] :
x ^ fintype.card G = 1
source
@[simp]
theorem card_nsmul_eq_zero {G : Type u_1} [add_group G] {x : G} [fintype G] :
fintype.card G • x = 0
source
theorem subgroup.pow_index_mem {G : Type u_1} [group G] (H : subgroup G) [H.normal] (g : G) :
g ^ H.index ∈ H
source
theorem add_subgroup.nsmul_index_mem {G : Type u_1} [add_group G] (H : add_subgroup G) [H.normal] (g : G) :
H.index • g ∈ H
source
theorem nsmul_eq_mod_card {G : Type u_1} [add_group G] {x : G} [fintype G] (n : ℕ) :
n • x = (n % fintype.card G) • x
source
theorem pow_eq_mod_card {G : Type u_1} [group G] {x : G} [fintype G] (n : ℕ) :
x ^ n = x ^ (n % fintype.card G)
source
theorem zpow_eq_mod_card {G : Type u_1} [group G] {x : G} [fintype G] (n : ℤ) :
x ^ n = x ^ (n % ↑(fintype.card G))
source
theorem zsmul_eq_mod_card {G : Type u_1} [add_group G] {x : G} [fintype G] (n : ℤ) :
n • x = (n % ↑(fintype.card G)) • x
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@[simp]
theorem pow_coprime_symm_apply {n : ℕ} {G : Type u_1} [group G] (h : (nat.card G).coprime n) (g : G) :
⇑((pow_coprime h).symm) g = g ^ (nat.card G).gcd_b n
source
noncomputable def pow_coprime {n : ℕ} {G : Type u_1} [group G] (h : (nat.card G).coprime n) :
G ≃ G

If gcd(|G|,n)=1 then the nth power map is a bijection

Equations
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@[simp]
theorem pow_coprime_apply {n : ℕ} {G : Type u_1} [group G] (h : (nat.card G).coprime n) (g : G) :
⇑(pow_coprime h) g = g ^ n
source
@[simp]
theorem nsmul_coprime_symm_apply {n : ℕ} {G : Type u_1} [add_group G] (h : (nat.card G).coprime n) (g : G) :
⇑((nsmul_coprime h).symm) g = (nat.card G).gcd_b n • g
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@[simp]
theorem nsmul_coprime_apply {n : ℕ} {G : Type u_1} [add_group G] (h : (nat.card G).coprime n) (g : G) :
⇑(nsmul_coprime h) g = n • g
source
noncomputable def nsmul_coprime {n : ℕ} {G : Type u_1} [add_group G] (h : (nat.card G).coprime n) :
G ≃ G

If gcd(|G|,n)=1 then the smul by n is a bijection

Equations
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@[simp]
theorem pow_coprime_one {n : ℕ} {G : Type u_1} [group G] (h : (nat.card G).coprime n) :
⇑(pow_coprime h) 1 = 1
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@[simp]
theorem nsmul_coprime_zero {n : ℕ} {G : Type u_1} [add_group G] (h : (nat.card G).coprime n) :
⇑(nsmul_coprime h) 0 = 0
source
@[simp]
theorem pow_coprime_inv {n : ℕ} {G : Type u_1} [group G] (h : (nat.card G).coprime n) {g : G} :
⇑(pow_coprime h) g⁻¹ = (⇑(pow_coprime h) g)⁻¹
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@[simp]
theorem nsmul_coprime_neg {n : ℕ} {G : Type u_1} [add_group G] (h : (nat.card G).coprime n) {g : G} :
⇑(nsmul_coprime h) (-g) = -⇑(nsmul_coprime h) g
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theorem inf_eq_bot_of_coprime {G : Type u_1} [group G] {H K : subgroup G} [fintype ↥H] [fintype ↥K] (h : (fintype.card ↥H).coprime (fintype.card ↥K)) :
H ⊓ K = ⊥
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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)) :
H ⊓ K = ⊥
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theorem image_range_add_order_of {G : Type u_1} [add_group G] {x : G} [fintype G] [decidable_eq G] :
finset.image (λ (i : ℕ), i • x) (finset.range (add_order_of x)) = ↑(add_subgroup.zmultiples x).to_finset
source
@[nolint]
theorem image_range_order_of {G : Type u_1} [group G] {x : G} [fintype 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.

source
theorem pow_gcd_card_eq_one_iff {G : Type u_1} [group G] {x : G} {n : ℕ} [fintype G] :
x ^ n = 1 ↔ x ^ n.gcd (fintype.card G) = 1

TODO: Generalise to finite + cancel_monoid.

source
theorem gcd_nsmul_card_eq_zero_iff {G : Type u_1} [add_group G] {x : G} {n : ℕ} [fintype G] :
n • x = 0 ↔ n.gcd (fintype.card G) • x = 0

TODO: Generalise to finite + cancel_add_monoid

source
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) :
add_submonoid M

A nonempty idempotent subset of a finite cancellative add monoid is a submonoid

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def submonoid_of_idempotent {M : Type u_1} [left_cancel_monoid M] [fintype M] (S : set M) (hS1 : S.nonempty) (hS2 : S * S = S) :
submonoid M

A nonempty idempotent subset of a finite cancellative monoid is a submonoid

Equations
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def add_subgroup_of_idempotent {G : Type u_1} [add_group G] [fintype G] (S : set G) (hS1 : S.nonempty) (hS2 : S + S = S) :
add_subgroup G

A nonempty idempotent subset of a finite add group is a subgroup

Equations
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def subgroup_of_idempotent {G : Type u_1} [group G] [fintype G] (S : set G) (hS1 : S.nonempty) (hS2 : S * S = S) :
subgroup G

A nonempty idempotent subset of a finite group is a subgroup

Equations
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@[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
source
def smul_card_add_subgroup {G : Type u_1} [add_group G] [fintype G] (S : set G) (hS : S.nonempty) :
add_subgroup G

If S is a nonempty subset of a finite add group G, then |G| • S is a subgroup

Equations
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@[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
source
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
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theorem order_of_abs_ne_one {G : Type u_1} [linear_ordered_ring G] {x : G} (h : |x| ≠ 1) :
order_of x = 0
source
theorem linear_ordered_ring.order_of_le_two {G : Type u_1} [linear_ordered_ring G] {x : G} :
order_of x ≤ 2
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@[protected]
theorem prod.order_of {α : Type u_4} {β : Type u_5} [monoid α] [monoid β] (x : α × β) :
order_of x = (order_of x.fst).lcm (order_of x.snd)
source
@[protected]
theorem prod.add_order_of {α : Type u_4} {β : Type u_5} [add_monoid α] [add_monoid β] (x : α × β) :
add_order_of x = (add_order_of x.fst).lcm (add_order_of x.snd)
source
theorem order_of_fst_dvd_order_of {α : Type u_4} {β : Type u_5} [monoid α] [monoid β] {x : α × β} :
order_of x.fst ∣ order_of x
source
theorem add_order_of_fst_dvd_add_order_of {α : Type u_4} {β : Type u_5} [add_monoid α] [add_monoid β] {x : α × β} :
add_order_of x.fst ∣ add_order_of x
source
theorem order_of_snd_dvd_order_of {α : Type u_4} {β : Type u_5} [monoid α] [monoid β] {x : α × β} :
order_of x.snd ∣ order_of x
source
theorem add_order_of_snd_dvd_add_order_of {α : Type u_4} {β : Type u_5} [add_monoid α] [add_monoid β] {x : α × β} :
add_order_of x.snd ∣ add_order_of x
source
theorem is_of_fin_order.fst {α : Type u_4} {β : Type u_5} [monoid α] [monoid β] {x : α × β} (hx : is_of_fin_order x) :
is_of_fin_order x.fst
source
theorem is_of_fin_add_order.fst {α : Type u_4} {β : Type u_5} [add_monoid α] [add_monoid β] {x : α × β} (hx : is_of_fin_add_order x) :
is_of_fin_add_order x.fst
source
theorem is_of_fin_add_order.snd {α : Type u_4} {β : Type u_5} [add_monoid α] [add_monoid β] {x : α × β} (hx : is_of_fin_add_order x) :
is_of_fin_add_order x.snd
source
theorem is_of_fin_order.snd {α : Type u_4} {β : Type u_5} [monoid α] [monoid β] {x : α × β} (hx : is_of_fin_order x) :
is_of_fin_order x.snd
source
theorem is_of_fin_order.prod_mk {α : Type u_4} {β : Type u_5} [monoid α] [monoid β] {a : α} {b : β} :
is_of_fin_order a → is_of_fin_order b → is_of_fin_order (a, b)
source
theorem is_of_fin_add_order.prod_mk {α : Type u_4} {β : Type u_5} [add_monoid α] [add_monoid β] {a : α} {b : β} :
is_of_fin_add_order a → is_of_fin_add_order b → is_of_fin_add_order (a, b)