They begin with symmetry groups
in various geometries, then describe an (apparently) different type of relationship between groups and geometry by introducing the fundamental group.
Being an important classification division in symmetry of crystals, each syngony includes several point symmetry groups
, Bravais lattices, and spatial symmetry groups
In this paper, we studied the symmetry groups
by using the classical Lie Group method to structural equation solutions.
The symmetry groups
admitted by the corresponding boundary value problem were obtained by using a special form of Lie group transformations, namely.
We will use orbifold notation to denote planar symmetry groups
in the sequel, compare .
DuVal  established that one only needs the cyclic groups 2Cn and 1Cn when considering the four discrete rotational symmetry groups
, i.e., the ones I am using for the quark families.
Other topics include molecules with torsional transitions of the exchange type, the meaning of the Born-Oppenheimer approximation, non-rigid molecular systems with continuous axial symmetry groups
, algebraic models of the global description of the molecular spectrum, and the Zeeman and Stark effects.
Although the modern theory of symmetry groups
and crystallography was not known in China we may notice that the creators of Chinese lattices used some principles that are similar to those that we can find in the papers of Weil, Coxeter, Conway (see ,,) and other geometers of the twentieth century.
His topics include some properties of the electromagnetic field near a rough metal surface, anomalies in the SER spectra of benzene adsorbed on lithium and hexafluorobenzene adsorbed on silver, electrodynamic forbiddance of the quadrupole enhancement mechanisms in molecules with cubic symmetry groups
, possible reasons for the competition of the Raman bands, and the charge transfer enhancement mechanism.
Then there exists a [GAMMA] equivariant homeomorphism germ ([R.sup.2n] x R, (0,0)) [right arrow] ([R.sup.2n] x R, (0,0)) which maps the nonlinear normal modes of H to those of [H.sub.2] + [H.sub.k], preserving their symmetry groups
. (Note that we do not assume time reversibility.