of Rosettes

In *S*^{2} and in *E*^{2} the 0-dimensional, point discrete symmetry
groups of rosettes *G*_{20} are the **cyclic groups****
C _{n}** (

*C _{n}* {

*D*_{n} {*S*_{1},*S*_{2}} *S*_{1}^{2} = *S*_{2}^{2} = (*S*_{1}*S*_{2})^{n} = *E*

*C*_{¥} {*S*_{1}}

*D*_{¥} {*S*_{1},*S*_{2}} *S*_{1}^{2} = *S*_{2}^{2} = *E*

All the symmetry groups **C _{n}** (

Presentation: {*S*} *S*^{n} = *E*

Order: *n* (*n* Î *N*)

Structure: *C*_{n}

Reducibility: If *n* = *km*, with (*k*,*m*) = 1,
then *C*_{n} = *C*_{k}×*C*_{m};
if *n* = *p*, with *p* - a prime number, then *C*_{n} 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*} *S*^{n} = *R*^{2} = (*SR*)^{2} = *E*

{*R*,*R*_{1}} *R*^{2} = *R*_{1}^{2} = (*RR*_{1})^{n} = *E* (*R*_{1} = *RS*)

Order: 2*n* (*n* Î *N*)

Structure: *D*_{n}

Reducibility: If *n* = 4*m*+2, then
*D*_{n} = *C*_{2}×D_{2m+1} = {*S*^{2m+1}} ×{*S*^{2},*R*} =

= {*Z*} ×{*S*^{2},*R*}; in other
cases *D*_{n} 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.

Enantiomorphism: there are no enantiomorphic modifications.

Polarity of rotations: rotations are non-polar.

Group-subgroup relations: [

[

The above survey of characteristics of the groups **
C _{n}** (

Cayley diagrams (Figure 2.1):

Cayley diagrams. |