Semi-dihedral group

The semi-dihedral (or quasi-dihedral) group SD₂ⁿ is one of the four 2-groups of order 2ⁿ (n ≥ 4) containing a cyclic maximal subgroup. It sits between the dihedral and generalized quaternion groups, with a twisted conjugation action that mixes features of both.

Definition

For n ≥ 4,

SD₂ⁿ = ⟨ r, s |  r2n−1 = s² = 1,  s r s⁻¹ = r2n−2 − 1 ⟩.

So r generates a cyclic subgroup of order 2n−1 (index 2), s is an involution, and conjugation by s raises r to r2n−2 − 1 — the "semi" exponent, neither −1 (dihedral) nor 1 + 2n−2 (the modular-maximal-cyclic case).

The six 2-groups of order 2ⁿ (n ≥ 4) with a cyclic index-2 subgroup are exactly:

Semi-dihedral exists only for n ≥ 4 — at order 8 the only choices are D₄ and Q₈, the two extraspecial 2-groups (no third twist yet).

Structure

For n ≥ 4, SD₂ⁿ has:

• Order 2ⁿ, nilpotency class n − 1 — maximal class for a 2-group of order 2ⁿ. The maximal-class 2-groups of order 2ⁿ are exactly D₂ⁿ, Q₂ⁿ, and SD₂ⁿ.

• Three maximal subgroups, all of order 2n−1:

  • the cyclic ⟨r⟩ ≅ C2n−1,
  • a dihedral D2n−1 (generated by s and r²),
  • a generalized quaternion Q2n−1 (generated by sr and r²).

  This three flavors in one is what the name "semi" alludes to.

• Center Z(SD₂ⁿ) = ⟨r2n−2⟩ ≅ C₂ — the unique element r2n−2 of order 2 inside ⟨r⟩.

• Commutator subgroup [SD₂ⁿ, SD₂ⁿ] = ⟨r²⟩ ≅ C2n−2.

• Involution count 1 + 2n−2: the central involution plus the 2n−2 even-power elements s·r2k in the s-coset. Compare: dihedral has 1 + 2n−1, generalized quaternion has just 1.

Examples

• SD₁₆ — the smallest, of order 16: ⟨r, s | r⁸ = s² = 1, s r s = r³⟩, SmallGroups id (16, 8). It contains C₈, D₄, and Q₈ as maximal subgroups.
• SD₃₂ — order 32, the unique maximal-class 2-group of that order besides D₃₂ and Q₃₂.
• Sylow 2-subgroup of GL(2,𝔽3) is SD₁₆. More generally, the Sylow 2-subgroup of GL(2,𝔽q) for q ≡ 3 (mod 4) is semi-dihedral.

Properties

• The semi-dihedral group is one of only six 2-groups with a cyclic maximal subgroup. For n ≥ 4, the 2-groups of order 2ⁿ with a cyclic subgroup of index 2 are exactly the cyclic C₂ⁿ, C2n−1 × C₂, dihedral, generalized quaternion, semi-dihedral, and modular maximal-cyclic groups.
• The semi-dihedral group SD₂ⁿ has maximal nilpotency class n − 1 — the largest possible for a 2-group of order 2ⁿ.
• The semi-dihedral group has three maximal subgroups, one of each flavor: SD₂ⁿ contains a cyclic, a dihedral, and a generalized quaternion maximal subgroup — the hybrid the name "semi" points to.
• The semi-dihedral group SD₂ⁿ has 1 + 2n−2 involutions: one central, plus 2n−2 on the dihedral side — between dihedral D₂ⁿ (1 + 2n−1) and generalized quaternion Q₂ⁿ (just 1).
• The Sylow 2-subgroup of GL(2,𝔽q) is semi-dihedral when q ≡ 3 (mod 4): the smallest case, GL(2,𝔽3), has Sylow 2-subgroup SD₁₆.
• The modular maximal-cyclic group is one of six 2-groups with a cyclic index-2 subgroup. For n ≥ 4, M₂ⁿ joins the cyclic C₂ⁿ, C2n−1 × C₂, dihedral D₂ⁿ, generalized quaternion Q₂ⁿ, and semi-dihedral SD₂ⁿ as the 2-groups of order 2ⁿ with a cyclic subgroup of index 2.

See also

• Dihedral group — the s r s⁻¹ = r⁻¹ cousin.
• Generalized quaternion group — the unique-involution cousin.
• Sylow subgroup — SD₂ⁿ arises as the Sylow 2-subgroup of GL(2,𝔽q) for q ≡ 3 (mod 4).
• Wikipedia: Semidihedral group