  An exposition of quantum permutation groups where an alternative to the
'Gelfand picture' of compact quantum groups is proposed. This point of view is
inspired by algebraic quantum mechanics and posits that states on the algebra
of continuous functions on a quantum permutation group can be interpreted as
quantum permutations. This interpretation allows talk of an element of a
compact quantum permutation group, and allows a clear understanding of the
difference between deterministic, random, and quantum permutations. The
interpretation is illustrated with the Kac-Paljutkin quantum group, the duals
of finite groups, as well as by other finite quantum group phenomena.
