In S2 and in E2 the 0-dimensional, point discrete symmetry groups of rosettes G20 are the cyclic groups Cn (n) and the dihedral groups Dn (nm) (n Î N). Also visually presentable is the continuous symmetry group of rosettes D¥ (¥m). Here, and in the sequel, we shall indicate the symbol of each symmetry group first according to G.E. Martin (1982), followed (in parentheses) by Shubnikov's notation (A.V. Shubnikov, V.A. Koptsik, 1974). By Cn (n), Dn (nm), C¥ (¥), D¥ (¥m) are denoted the symmetry groups of rosettes G20, distinct from the abstract groups, denoted by Cn, Dn, C¥, D¥, and given by the presentations:
Cn {S1} S1n = E
Dn {S1,S2} S12 = S22 = (S1S2)n = E
C¥ {S1}
D¥ {S1,S2} S12 = S22 = E
All the symmetry groups Cn (n) or Dn
(nm), obtained for different values of n (n
Î N) are
called the symmetry groups of the type Cn (n) or
Dn (nm).
Presentation: {S} Sn = E
Order: n (n Î N)
Structure: Cn
Reducibility: If n = km, with (k,m) = 1,
then Cn = Ck×Cm;
if n = p, with p - a prime number, then Cn is
irreducible.
Form of the fundamental region: unbounded, allows variation of the
shape of its boundaries.
Enantiomorphism: enantiomorphic modifications exist.
Polarity of rotations: rotations are polar.
Presentations: {S,R} Sn = R2 = (SR)2 = E
{R,R1} R2 = R12 = (RR1)n = E (R1 = RS)
Order: 2n (n Î N)
Structure: Dn
Reducibility: If n = 4m+2, then
Dn = C2×D2m+1 = {S2m+1} ×{S2,R} =
Form of the fundamental region: unbounded, of a fixed shape, with
rectilinear boundaries.
Enantiomorphism: there are no enantiomorphic modifications.
Polarity of rotations: rotations are non-polar.
Enantiomorphism: there are no enantiomorphic modifications.
Polarity of rotations: rotations are non-polar.
The above survey of characteristics of the groups
Cn (n) and Dn (nm) is based on
I. Grossman, W. Magnus (1964), W. Magnus, A. Karras, S. Solitar
(1966), L.C. Biedenharn, W. Brouver, W.T. Sharp (1968),
A.V. Shubnikov, V.A. Koptsik (1974), H.S.M. Coxeter, W.O.J. Moser
(1980).
Cayley diagrams (Figure 2.1):
Cayley diagrams.
Cn
(n)
Dn
(nm)
= {Z} ×{S2,R}; in other
cases Dn is irreducible.
D¥
(¥m)
Group-subgroup relations: [Dn:Cn] = 2 [
Dkm:Dm] = k
(in particular
[D2m:D2] = m)
[Ckm:Cm] = k
(in particular [C2m:C2] = m)
NEXT
CONTENTS