Heisenberg groups

The Heisenberg group H(m, R) is the group of unitriangular matrices over a ring R whose only nonzero entries off the diagonal lie in the first row and the last column:

[ 1  a₁ … a_k  c  ]
[ 0  1      0  b₁ ]
[ ⋮         ⋱   ⋮ ]
[ 0  0      1  b_k ]
[ 0  0      0  1  ]

with m = 2k + 1 free entries a₁…a_k, b₁…b_k, c in an (k+2) × (k+2) matrix. Writing an element as the triple (a, b, c), the product is

(a, b, c)(a′, b′, c′) = (a + a′, b + b′, c + c′ + a·b′).

Order — |H(m, R)| = |R|^m, one ring entry per free position.
Nilpotent of class 2 — the center and the derived subgroup coincide, both equal to the corner {(0, 0, c)} ≅ (R, +).
2-step, always — unlike UT(n,𝔽_q), whose class grows with n, every Heisenberg group has class exactly 2.

Relation to the unitriangular groups

At m = 3 the two shapes are the same: in a 3 × 3 matrix "first row and last column" is every position above the diagonal, so

H(3, R) = UT(3, R).

Above that Heisenberg is a proper subgroup — H(5,𝔽_q) sits inside UT(4,𝔽_q) with the (2,3) entry forced to zero, of index q (order q⁵ against q⁶).

Over a prime field: the extraspecial groups

Over 𝔽_p the center is (𝔽ₚ, +) ≅ Cₚ, so H(m,𝔽_p) is extraspecial of order p^m — and always of plus type:

H(2k+1,𝔽_p) ≅ p₊^1+2k

For odd p the power law (a,b,c)ⁿ = (na, nb, nc + C(n,2)·a·b) gives exponent p, since p divides C(p, 2).
For p = 2 squaring gives the form q(a, b) = a·b on G/Z ≅ 𝔽₂^2k, hyperbolic of Witt index k. The smallest case is H(3,𝔽₂) = D₄, and H(5,𝔽₂) = D₄ ∘ D₄.

The minus type — exponent p² for odd p, the factor Q₈ for p = 2 — is never a Heisenberg group.

Over other rings

Past the prime fields the center grows and the group stops being extraspecial, though it stays special (center = derived subgroup = Frattini subgroup, elementary abelian):

Group	Order	Center
`H(3,𝔽₂)`	8	`C₂` — extraspecial `2₊¹⁺²`, the group `D₄`
`H(3,𝔽₄)`	64	`C₂ × C₂` — special, exponent 4
`H(3,ℤ/4)`	64	`C₄` — not even special

H(3,𝔽₄) and H(3,ℤ/4) are the smallest illustration that the ring, not just its size, decides the group: same order, different centers, and so non-isomorphic. The first is 4-generated of exponent 4; the second is 2-generated of exponent 8.

Elsewhere

The discrete Heisenberg group H(3, ℤ) is the free class-2 nilpotent group on two generators; H(3, Z/d) is its reduction mod d, known in quantum information as the Weyl–Heisenberg group (the qudit Pauli group, up to a phase convention that differs for even d).

Part of the wider family of matrix groups.

Wikipedia: Heisenberg group