Dicyclic groups

The dicyclic group Dicₙ (order 4n, also written Q₄ₙ) is the non-split cyclic-by-C₂ extension in which a second generator inverts the cyclic part and squares to the unique central involution. Its 2-power members are the generalized quaternion groups.

Definition

For n ≥ 1,

Dicₙ = ⟨ a, b |  a²ⁿ = 1,  b² = aⁿ,  b⁻¹ a b = a⁻¹ ⟩,    |Dicₙ| = 4n.

⟨a⟩ ≅ C₂ₙ is cyclic of index 2; every element outside it has the form aᵏb. Also written Q₄ₙ, so Dic₂ = Q₈, Dic₃ = Q₁₂, Dic₄ = Q₁₆.

Structure

• Cyclic index-2 subgroup ⟨a⟩ ≅ C₂ₙ, normal.
• A unique element of order 2, namely aⁿ. Every element of Dicₙ ∖ ⟨a⟩ satisfies (aᵏb)² = aⁿ, so it has order 4; and aⁿ is the only involution inside ⟨a⟩. Thus Dicₙ has exactly one involution.
• Center Z(Dicₙ) = ⟨aⁿ⟩ ≅ C₂ for n ≥ 2 (the degenerate Dic₁ is abelian). The central involution aⁿ = b².
• Every abelian subgroup is cyclic. With a single involution, each Sylow 2-subgroup is cyclic or generalized quaternion — each with a unique involution — and every Sylow p-subgroup for odd p lies in the cyclic ⟨a⟩; an abelian group with cyclic Sylow subgroups is cyclic.
• Metacyclic (cyclic normal ⟨a⟩, cyclic quotient C₂), hence supersolvable and solvable.
• Nilpotent ⟺ n is a power of 2 ⟺ Dicₙ is a 2-group; for n ≥ 2 these are exactly the generalized quaternion groups.

Split vs. non-split: the dihedral cousin

The extension

1 → C₂ₙ → Dicₙ → C₂ → 1

is a non-split extension of C₂ by C₂ₙ: since the only involution aⁿ lies in ⟨a⟩, the quotient generator has no order-2 lift, so ⟨a⟩ has no complement. The dihedral group D₂ₙ uses the same inversion action b⁻¹ a b = a⁻¹ but closes with b² = 1, making it the split extension C₂ₙ ⋊ C₂ — a semidirect product.

Generalized quaternion case

When n = 2ᵏ⁻² (k ≥ 3), Dicₙ has order 2ᵏ and is the generalized quaternion group Q₂ᵏ. These are exactly the non-abelian nilpotent dicyclic groups — equivalently, the non-cyclic 2-groups with a unique involution.

Dicₙ is also the binary dihedral group: the preimage of the order-2n dihedral rotation group Dₙ ⊂ SO(3) under the double cover SU(2) → SO(3), realizing Dicₙ as a finite subgroup of the unit quaternions — the source of the Q notation.

Examples

• Dic₁ — order 4, degenerate: b² = a and a is inverted trivially, so the group is abelian, Dic₁ ≅ C₄. Usually excluded.
• Dic₂ = Q₈ — order 8, the quaternion group {±1, ±i, ±j, ±k} (a = i, b = j); one of the two non-abelian groups of order 8, alongside D₄.
• Dic₃ = Q₁₂ — order 12, the smallest dicyclic group that is not a 2-group: non-nilpotent, metacyclic. One of three non-abelian groups of order 12 (with A₄ and D₆).
• Dic₄ = Q₁₆ — order 16, generalized quaternion.

See also

• Generalized quaternion group — the 2-power dicyclic groups.
• Dihedral group — the split cousin with the same inversion action.
• Metacyclic group, cyclic group.
• Extension, semidirect product — non-split vs. split.
• Wikipedia: Dicyclic group