Abstract A quaternionic Hadamard matrix (QHM) of ordernis an $$n\times n$$ n×n matrixHwith non-zero entries in the quaternions such that $$HH^*=nI_n$$ HH∗=nIn , where $$I_n$$ In and $$H^*$$ H∗ denote the identity matrix and the conjugate-transpose ofH, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.

We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.

Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of plain groups, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in ΣP3. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).

We show that there is a dense set of group orders such that for every such order we can decide in nearly-linear time whether two multiplication tables describe isomorphic groups. This improves significantly over the general quasi-polynomial time complexity and shows that group isomorphism can be tested efficiently for almost all group orders. We also show that in nearly-linear time it can be decided whether a multiplication table describes a group; this improves over the known super-linear complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model, but we give the implications to a RAM model in the promise hierarchy as well.