There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG($n, p$), $n$ being odd and $p
$ a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of $n$-simplices, i. e., pairs of $n$-simplices which are emph{mutually inscribed and circumscribed} to each other. For group elements representing an $n$-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension $n$ of the associated polar space in group theoretic terms. Any Moebius pair of $n$-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for $p=2$ and $n=5$.
Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessar
y and sufficient conditions to be met in order to carry out the following programme: Given a group $vG$, we first construct vector spaces over $GF(p)$, $p$ a prime, by factorising $vG$ over appropriate normal subgroups. Then, by expressing $GF(p)$ in terms of the commutator subgroup of $vG$, we construct alternating bilinear forms, which reflect whether or not two elements of $vG$ commute. Restricting to $p=2$, we search for ``refinements in terms of quadratic forms, which capture the fact whether or not the order of an element of $vG$ is $leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation of several distinct elements of $vG$. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.
