  We discuss generalizations of the notion of i) the group of unitary elements
of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations
and iii) (real) automorphisms, in the framework of compact quantum group theory
and spectral triples. The quantum analogue of these groups are defined as
universal (initial) objects in some natural categories. After proving the
existence of the universal objects, we discuss several examples that are of
interest to physics, as they appear in the noncommutative geometry approach to
particle physics: in particular, the C*-algebras M_n(R), M_n(C) and M_n(H),
describing the finite noncommutative space of the Einstein-Yang-Mills systems,
and the algebras A_F=C+H+M_3(C) and A^{ev}=H+H+M_4(C), that appear in
Chamseddine-Connes derivation of the Standard Model of particle physics
minimally coupled to gravity. As a byproduct, we identify a "free" version of
the symplectic group Sp(n) (quaternionic unitary group).
