Generalized quaternion groups

The generalized quaternion group Q₂ⁿ (n ≥ 3) is the dicyclic group of 2-power order 2ⁿ: the non-split cyclic-by-C₂ extension in which C₂ acts by inversion and the complementary generator squares to the unique central involution rather than to 1 (so it has order 4, not 2). The classical quaternion group Q₈ = {±1, ±i, ±j, ±k} is the case n = 3.

Definition

For n ≥ 3,

Q₂ⁿ = ⟨ a, b |  a^2ⁿ⁻¹ = 1,  b² = a^2ⁿ⁻²,  b⁻¹ a b = a⁻¹ ⟩,    |Q₂ⁿ| = 2ⁿ.

a generates a cyclic subgroup ⟨a⟩ ≅ C₂ⁿ⁻¹ of index 2, normal in Q₂ⁿ.
b inverts a, and b² = a^2ⁿ⁻² is the unique involution — so b has order 4 and the extension does not split.
In the dicyclic naming Dicₘ (order 4m), Q₂ⁿ = Dic₂ⁿ⁻²; e.g. Q₈ = Dic₂.

Q₈ (n = 3): taking a = i, b = j gives a⁴ = 1, b² = a² = −1, b⁻¹ a b = a⁻¹, the unit quaternions {±1, ±i, ±j, ±k} with ab = ij = k.

Structure

Unique involution. Q₂ⁿ has exactly one element of order 2, a^2ⁿ⁻² = b². It is central and generates Z(Q₂ⁿ) ≅ C₂.
Abelian subgroups are cyclic. Every abelian subgroup of Q₂ⁿ is cyclic; the largest is ⟨a⟩ ≅ C₂ⁿ⁻¹ of index 2. More strongly, every subgroup is cyclic or generalized quaternion (each nontrivial subgroup contains the one involution).
Maximal class. Nilpotency class n − 1, the maximum for a group of order 2ⁿ. The 2-groups of maximal class are exactly the dihedral D₂ⁿ, semi-dihedral SD₂ⁿ (n ≥ 4), and generalized quaternion Q₂ⁿ.
Metacyclic (cyclic normal ⟨a⟩ with cyclic quotient C₂), hence supersolvable and solvable; nilpotent as a 2-group.
Maximal subgroups (n ≥ 4): three, each of order 2ⁿ⁻¹ — the cyclic ⟨a⟩ ≅ C₂ⁿ⁻¹ and two copies of Q₂ⁿ⁻¹. (For Q₈ the three maximal subgroups ⟨i⟩, ⟨j⟩, ⟨k⟩ are all C₄.)

`Q₈` in particular

Smallest Hamiltonian group: non-abelian, yet every subgroup is normal. The larger Q₂ⁿ (n ≥ 4) are not Hamiltonian — e.g. ⟨b⟩ is non-normal in Q₁₆. By the Dedekind–Baer theorem every finite Hamiltonian group is Q₈ × B × A with B elementary abelian 2-group and A abelian of odd order.
Extraspecial of order 8, the minus type 2¹⁺²₋; the dihedral D₄ is the plus type 2¹⁺²₊. These are the only two non-abelian groups of order 8. (Q₂ⁿ is extraspecial only for n = 3.)
Q₈ / Z(Q₈) ≅ C₂ × C₂.

Characterization

For a finite p-group P, the following are equivalent:

P has a unique subgroup of order p;
every abelian subgroup of P is cyclic.

Such a P is cyclic (any p), or p = 2 and P is generalized quaternion.

The generalized quaternion groups are exactly the noncyclic finite p-groups with a single subgroup of order p — equivalently, the noncyclic finite p-groups whose abelian subgroups are all cyclic. Either condition forces p = 2: a single involution.

For odd p, a unique subgroup of order p forces P cyclic; generalized quaternion is the only noncyclic possibility, and only at p = 2.

Occurrence

Sylow 2-subgroups of SL(2,𝔽_q), q odd. SL(2,𝔽_q) has a unique involution −I, so its Sylow 2-subgroups have a unique involution and are generalized quaternion. Smallest case: SL(2,𝔽₃) (order 24) has Sylow 2-subgroup Q₈; SL(2,𝔽₅) (order 120) also has Q₈, and SL(2,𝔽₇) (order 336) has Q₁₆.

Examples

Q₈ — order 8, the quaternion units {±1, ±i, ±j, ±k}. Smallest Hamiltonian group; extraspecial 2¹⁺²₋.
Q₁₆ — order 16: ⟨a, b | a⁸ = 1, b² = a⁴, b⁻¹ a b = a⁻¹⟩. Maximal subgroups C₈ and two Q₈.
Q₃₂ — order 32, one of the three maximal-class 2-groups of that order (with D₃₂, SD₃₂).