The discrete Heisenberg group and its automorphism group

Skew braces and Hopf–Galois structures of Heisenberg type

“…The next proposition determines the automorphism group of M 1 . For the analogous result over Z see [25]. I am grateful to the referee for drawing my attention to this reference.…”

Representations of the discrete Heisenberg group on distribution spaces of two-dimensional local fields

“…Besides an action of the element u is u(µ B,r,s ) = q r−s µ B,r,s (we recall that the pair t, u is the system of local parameters of K ). Now from an explicit description of the group K * given above, by a direct calculation we obtain that the composition of map (8) with the quotient map K * → K * /O ′ * is an isomorphism between groups G and K * /O ′ * . The proposition is proved.…”

“…The first named author proved in [8] that this homomorphism can be extended to an injective homomorphism GL(2, Z) ֒→ Aut(G) . By means of the last homomorphism it was also proved in [8] that the group Aut(G) is isomorphic to (Z ⊕ Z) ⋊ GL(2, Z) . We note that other isomorphism Aut(G) ≃ (Z ⊕ Z) ⋊ GL(2, Z) was constructed earlier by P. Kahn in [5] by another methods.…”

Conjugacy growth in the higher Heisenberg groups

“…Remark 3.14. Although we do not need it for the arguments that follow, we note that in the special case of N = H r , we can use a straightforward generalisation of an argument of Osipov [27] to extend Proposition 3.13 to give the following short exact sequence, where Inn(H r ) = Z 2r :…”