## Chapter 1.6

###
Symbols of Symmetry Groups

When denoting symmetry groups and their generalizations,
antisymmetry and colored symmetry groups, we always come across
the unpleasant task of trying, at least to some extent, to
reconcile and bring to accord the different sources and symbols
used in literature. Most of symbols come from the work of
crystallographers, some from the mathematicians who were
engaged in studies of the theory of symmetry, while some chapters
(e.g., that on conformal symmetry) demand the introduction of new
symbols. Since only lately there have been attempts to make
uniform the symbols of symmetry groups, positive results are
mainly achieved with the symmetry groups of ornaments *G*_{2}
(International symbols). In the other cases, a great number of
authors, with their original results introduced together new or
modified symbols. Therefore, it is unavoidable to accept the
compromise solution and quote several alternative kinds of
symbols. Also, this offers possibilities for the application of
optimal symbols in each particular case, since for the different
practical needs of the theory of symmetry, every kind of symbols
has its advantages, but also, disadvantages.

For denoting the symmetry groups of friezes and
ornaments, the simplified version of the International symbols
(M. Senechal, 1975;
H.S.M. Coxeter, W.O.J. Moser, 1980)
will be used, while in other cases the non-coordinate symbols, used by
Soviet authors (A.V. Shubnikov, V.A. Koptsik, 1974) will be
indicated also. The **symbols of antisymmetry and colored
symmetry groups** will be given in the group/subgroup notation
(*G*/*H*, *G*/*H*/*H*_{1})
(A.V. Shubnikov, V.A. Koptsik, 1974;
A.M. Zamorzaev, 1976;
H.S.M. Coxeter, 1985, 1987;
V.A. Koptsik,
J.N. Kotzev, 1974).

The **International symbols** are coordinate symbols
of symmetry groups. For the symmetry groups of friezes and
ornaments, the first coordinate denotes the translational
subgroup **p** (**c** with the rhombic lattice) while the
other coordinates are symbols of glide reflections **g** and
reflections **m** perpendicular to the corresponding coordinate
axis and symbols of the rotation axis **n** collinear with the
corresponding coordinate axis.

The **non-coordinate symbols** of symmetry groups are
mainly used in the works of Soviet authors, in which (**a**)
denotes a translation, (**ã**) a glide reflection, **
n** the order of a rotation, the absence of symbols between
elements - collinearity (incidence) of relevant elements of
symmetry (denoted in the original works by the symbol
·), while the symbol **:** denotes perpendicularity of relevant
symmetry elements. For example, the symmetry groups **D**_{4}
and **C**_{4} will be denoted, respectively, by **4** and
**4m**.

Given at the beginning of each chapter is a survey of the
geometric-algebraic characteristics of the groups of symmetry
discussed: presentation, group order, group structure,
reducibility, form of the fundamental region, enantiomorphism,
polarity (non-polarity, bipolarity), group-subgroup relations,
table of minimal indexes of subgroups in groups, Cayley diagrams.
Further on are discussed the antisymmetry and color-symmetry
desymmetrizations, construction methods, questions related to
continuous groups and to different problems of
algebraic-geometric properties of symmetry groups, which
directly influence the different visual parameters.