Modular maximal-cyclic group

The modular maximal-cyclic 2-group M₂ⁿ (often just called the modular group of order 2ⁿ) is the quietest of the six 2-groups of order 2ⁿ (n ≥ 4) containing a cyclic subgroup of index 2 — the others being the cyclic C₂ⁿ, the abelian C2n−1 × C₂, dihedral, generalized quaternion, and semi-dihedral. Unlike the last three, M₂ⁿ is not of maximal class: its nilpotency class is just 2.

Not to be confused with the modular group PSL₂(ℤ) of complex analysis. The name here comes from its modular subgroup lattice (its subgroups all permute), not from modular forms.

Definition

For n ≥ 4,

M₂ⁿ = ⟨ r, s |  r2n−1 = s² = 1,  s r s⁻¹ = r1 + 2n−2 ⟩.

So r generates a cyclic subgroup of order 2n−1 (index 2), and s twists it by a small exponent 1 + 2n−2 — close to the identity rather than to −1 (dihedral) or 2n−2 − 1 (semi-dihedral). At n = 3 the recipe collapses: 1 + 2 = 3 ≡ −1 (mod 4), so M₈ = D₄. The family only starts to differ at n = 4.

Structure

• Order 2ⁿ, nilpotency class 2 (not maximal — D₂ⁿ, Q₂ⁿ, SD₂ⁿ all have class n − 1).
• Metacyclic and supersolvable: the split semidirect product C2n−1 ⋊ C₂, a cyclic normal subgroup ⟨r⟩ with cyclic quotient ⟨s⟩.
• Center Z(M₂ⁿ) = ⟨r²⟩ ≅ C2n−2 — a large center, in contrast to the order-2 centers of D / Q / SD.
• Commutator subgroup [M₂ⁿ, M₂ⁿ] = ⟨r2n−2⟩ ≅ C₂, so the abelianization is M₂ⁿ / [M₂ⁿ, M₂ⁿ] ≅ C2n−2 × C₂.
• Exactly 3 involutions: r2n−2, s, and s · r2n−2. (Compare D₂ⁿ: 1 + 2n−1; Q₂ⁿ: 1; SD₂ⁿ: 1 + 2n−2.)
• Exponent 2n−1 — the element r (and many of its products with s) achieves the maximum possible exponent for a 2-group of order 2ⁿ that isn't cyclic.
• Modular subgroup lattice: every subgroup is permutable (HK = KH). This is the property the name "modular" refers to.

Examples

• M₁₆ — the smallest, of order 16: ⟨r, s | r⁸ = s² = 1, s r s = r⁵⟩, SmallGroups id (16, 6). Three involutions: r⁴, s, s · r⁴. Center ⟨r²⟩ ≅ C₄.
• M₃₂, M₆₄, … — exist at every order 2ⁿ for n ≥ 4, one iso-type each.

Properties

• 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.
• The modular maximal-cyclic group is the only nilpotency-class-2 member among the 2-groups of order 2ⁿ with a cyclic index-2 subgroup: M₂ⁿ has class 2, while the dihedral, generalized quaternion, and semi-dihedral groups of that order have maximal class n − 1.
• The modular maximal-cyclic group has a large center: Z(M₂ⁿ) is cyclic of order 2n−2 (the squares in its cyclic maximal subgroup), versus the order-2 center of its maximal-class siblings.
• The modular maximal-cyclic group M₂ⁿ has exactly three involutions for every n ≥ 4 — a constant, unlike the growing dihedral (1 + 2n−1) and semi-dihedral (1 + 2n−2) counts. It shares this count with C₂ × Q2n−1, but is set apart by having an element of order 2n−1 (a cyclic subgroup of index 2).
• The modular maximal-cyclic group has a modular subgroup lattice: every subgroup permutes with every other (HK = KH), the property that makes its subgroup lattice modular and gives the family its name.

See also

• Cyclic group, dihedral group, generalized quaternion group, semi-dihedral group — the four sibling 2-groups with a cyclic index-2 subgroup.
• Metacyclic group — every one of these siblings is metacyclic.
• Wikipedia: Modular maximal-cyclic group