  Finite group symmetry is commonplace in Physics, in particular through
crystallographic groups occurring in condensed matter physics -- but also
through the inversions (C,P,T and their combinations) occurring in high energy
physics and field theory.
  The breaking of finite groups symmetry has thus been thoroughly studied, and
general approaches exist to investigate it.
  In Landau theory, the state of a system is described by a finite dimensional
variable (the {\it order parameter}), and physical states correspond to minima
of a potential, invariant under a group.
  In this article we describe the basics of symmetry breaking analysis for
systems described by a symmetric polynomial; in particular we discuss generic
symmetry breakings, i.e. those determined by the symmetry properties themselves
and independent on the details of the polynomial describing a concrete system.
We also discuss how the plethora of invariant polynomials can be to some extent
reduced by means of changes of coordinates, i.e. how one can reduce to consider
certain types of polynomials with no loss of generality. Finally, we will give
some indications on extension of this theory, i.e. on how one deals with
symmetry breakings for more general groups and/or more general physical
systems.
