Symmetry Groups    of Rosettes G20

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).

Cn    (n)

Presentation: {S}     Sn = E

Order: n (n Î N)

Structure: Cn

Form of the fundamental region: unbounded, allows variation of the shape of its boundaries.

Enantiomorphism: enantiomorphic modifications exist.

Polarity of rotations: rotations are polar.

Dn    (nm)

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} =
= {Z} ×{S2,R}; in other cases Dn is irreducible.

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.

D¥    (¥m)

Enantiomorphism: there are no enantiomorphic modifications.

Polarity of rotations: rotations are non-polar.

Group-subgroup relations: [Dn:Cn] = 2 [ Dkm:Dm] = k (in particular
[D2m:D2] = m) [Ckm:Cm] = k (in particular [C2m:C2] = m)

Cayley diagrams (Figure 2.1):

Figure 2.1
 Cayley diagrams.