Documentation

Mathlib.GroupTheory.SpecificGroups.Quaternion

Quaternion Groups #

We define the (generalised) quaternion groups QuaternionGroup n of order 4n, also known as dicyclic groups, with elements a i and xa i for i : ZMod n. The (generalised) quaternion groups can be defined by the presentation $\langle a, x | a^{2n} = 1, x^2 = a^n, x^{-1}ax=a^{-1}\rangle$. We write a i for $a^i$ and xa i for $x * a^i$. For n=2 the quaternion group QuaternionGroup 2 is isomorphic to the unit integral quaternions (Quaternion ℤ)ˣ.

Main definition #

QuaternionGroup n: The (generalised) quaternion group of order 4n.

Implementation notes #

This file is heavily based on DihedralGroup by Shing Tak Lam.

In mathematics, the name "quaternion group" is reserved for the cases n ≥ 2. Since it would be inconvenient to carry around this condition we define QuaternionGroup also for n = 0 and n = 1. QuaternionGroup 0 is isomorphic to the infinite dihedral group, while QuaternionGroup 1 is isomorphic to a cyclic group of order 4.

References #

TODO #

Show that QuaternionGroup 2 ≃* (Quaternion ℤ)ˣ.

source
inductive QuaternionGroup (n : ℕ) :
Type

The (generalised) quaternion group QuaternionGroup n of order 4n. It can be defined by the presentation $\langle a, x | a^{2n} = 1, x^2 = a^n, x^{-1}ax=a^{-1}\rangle$. We write a i for $a^i$ and xa i for $x * a^i$.

Instances For
    source
    instance instDecidableEqQuaternionGroup :
    {n : ℕ} → DecidableEq (QuaternionGroup n)
    Equations
    • instDecidableEqQuaternionGroup = decEqQuaternionGroup✝
    source
    instance QuaternionGroup.instInhabited {n : ℕ} :
    Inhabited (QuaternionGroup n)
    Equations
    • QuaternionGroup.instInhabited = { default := QuaternionGroup.one }
    source
    instance QuaternionGroup.instGroup {n : ℕ} :
    Group (QuaternionGroup n)

    The group structure on QuaternionGroup n.

    Equations
    source
    @[simp]
    theorem QuaternionGroup.a_mul_a {n : ℕ} (i : ZMod (2 * n)) (j : ZMod (2 * n)) :
    QuaternionGroup.a i * QuaternionGroup.a j = QuaternionGroup.a (i + j)
    source
    @[simp]
    theorem QuaternionGroup.a_mul_xa {n : ℕ} (i : ZMod (2 * n)) (j : ZMod (2 * n)) :
    QuaternionGroup.a i * QuaternionGroup.xa j = QuaternionGroup.xa (j - i)
    source
    @[simp]
    theorem QuaternionGroup.xa_mul_a {n : ℕ} (i : ZMod (2 * n)) (j : ZMod (2 * n)) :
    QuaternionGroup.xa i * QuaternionGroup.a j = QuaternionGroup.xa (i + j)
    source
    @[simp]
    theorem QuaternionGroup.xa_mul_xa {n : ℕ} (i : ZMod (2 * n)) (j : ZMod (2 * n)) :
    QuaternionGroup.xa i * QuaternionGroup.xa j = QuaternionGroup.a (↑n + j - i)
    source
    theorem QuaternionGroup.one_def {n : ℕ} :
    1 = QuaternionGroup.a 0
    source
    def QuaternionGroup.quaternionGroupZeroEquivDihedralGroupZero :
    QuaternionGroup 0 ≃* DihedralGroup 0

    The special case that more or less by definition QuaternionGroup 0 is isomorphic to the infinite dihedral group.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      source
      instance QuaternionGroup.instFintypeOfNeZeroNat {n : ℕ} [NeZero n] :
      Fintype (QuaternionGroup n)

      If 0 < n, then QuaternionGroup n is a finite group.

      Equations
      source
      instance QuaternionGroup.instNontrivial {n : ℕ} :
      Nontrivial (QuaternionGroup n)
      Equations
      • ⋯ = ⋯
      source
      theorem QuaternionGroup.card {n : ℕ} [NeZero n] :
      Fintype.card (QuaternionGroup n) = 4 * n

      If 0 < n, then QuaternionGroup n has 4n elements.

      source
      @[simp]
      theorem QuaternionGroup.a_one_pow {n : ℕ} (k : ℕ) :
      QuaternionGroup.a 1 ^ k = QuaternionGroup.a ↑k
      source
      theorem QuaternionGroup.a_one_pow_n {n : ℕ} :
      QuaternionGroup.a 1 ^ (2 * n) = 1
      source
      @[simp]
      theorem QuaternionGroup.xa_sq {n : ℕ} (i : ZMod (2 * n)) :
      QuaternionGroup.xa i ^ 2 = QuaternionGroup.a ↑n
      source
      @[simp]
      theorem QuaternionGroup.xa_pow_four {n : ℕ} (i : ZMod (2 * n)) :
      QuaternionGroup.xa i ^ 4 = 1
      source
      @[simp]
      theorem QuaternionGroup.orderOf_xa {n : ℕ} [NeZero n] (i : ZMod (2 * n)) :
      orderOf (QuaternionGroup.xa i) = 4

      If 0 < n, then xa i has order 4.

      source
      theorem QuaternionGroup.quaternionGroup_one_isCyclic :
      IsCyclic (QuaternionGroup 1)

      In the special case n = 1, Quaternion 1 is a cyclic group (of order 4).

      source
      @[simp]
      theorem QuaternionGroup.orderOf_a_one {n : ℕ} :
      orderOf (QuaternionGroup.a 1) = 2 * n

      If 0 < n, then a 1 has order 2 * n.

      source
      theorem QuaternionGroup.orderOf_a {n : ℕ} [NeZero n] (i : ZMod (2 * n)) :
      orderOf (QuaternionGroup.a i) = 2 * n / (2 * n).gcd i.val

      If 0 < n, then a i has order (2 * n) / gcd (2 * n) i.

      source
      theorem QuaternionGroup.exponent {n : ℕ} :
      Monoid.exponent (QuaternionGroup n) = 2 * lcm n 2