  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 $n=5$.
