In the plane S^{2} and E^{2} there are seven discrete, onedimensional, line groups of isometries, the symmetry groups of friezes G_{21}: p11 (11), p1g (1g), p12 (12), pm1 (m1), p1m (1m), pmg (mg), pmm (mm) and two visually presentable continuous symmetry groups of friezes: p_{0}m1=pm_{0}1 (m_{0}1) and p_{0}mm= pm_{0}m (m_{0}m).
To denote them, we have used the simplified version of International Twodimensional Symbols (M. Senechal, 1975; H.S.M. Coxeter, 1985). Here, the first symbol represents an element of symmetry perpendicular to the direction of the translation, while the second denotes an element of symmetry parallel or perpendicular (exclusively for 2rotations) to the direction of the translation.
Presentations and structures:
11  {X}  C_{¥}  
1g  {P}  C_{¥}  
12  {X,T}  T^{2} = (TX)^{2} = E  D_{¥}  
{T,T_{1}}  T^{2} = T_{1}^{2} = E  (T_{1} = TX)  
m1  {X,R_{1}}  R_{1}^{2} = (R_{1}X)^{2} = E  D_{¥}  
{R_{1},R_{2}}  R_{1}^{2} = R_{2}^{2} = E  (R_{2} = R_{1}X)  
1m  {X,R}  R^{2} = E  RX = XR  C_{¥} ×D_{1}  
mg  {P,R_{1}}  R_{1}^{2} = (R_{1}P)^{2} = E  D_{¥}  
{R_{1},T}  R_{1}^{2} = T^{2} = E  (T = R_{1}P)  
mm  {X,R,R_{1}}  R^{2} = R_{1}^{2} = (R_{1}X)^{2} = E  RX = XR  RR_{1} = R_{1}R  D_{¥} ×D_{1}  
{R,R_{1},R_{2}}  R^{2} = R_{1}^{2} = R_{2}^{2} = E  RR_{1} = R_{1}R  RR_{2} = R_{2}R  (R_{2} = R_{1}X)  
Form of the fundamental region: unbounded, allows variation of the boundaries that do not belong to reflection lines.
Enantiomorphism: 11, 12 possess enantiomorphic modifications, while in other cases the enantiomorphism does not occur.
Polarity of rotations: polar rotations  12, mg;
nonpolar rotations  mm, m_{0}m.
Polarity of translations: polar translations 
11, 1g, 1m;
bipolar translations  12;
nonpolar translations  m1, mg, mm,
m_{0}1, m_{0}m.
The table of minimal indexes of subgroups in groups:
11  1g  12  m1  1m  mg  mm  
11  2  
1g  2  3  
12  2  2  
m1  2  2  
1m  2  2  2  
mg  4  2  2  2  3  
mm  4  4  2  2  2  2  2 
All the discrete symmetry groups of friezes are subgroups of the group mm generated by reflections and given by the presentation:

R_{1}, T = RR_{2}  generate  mg  D_{¥}  
R, X = R_{1}R_{2}  generate  1m  C_{¥} ×D_{1}  
R_{1}, X  generate  m1  D_{¥}  
X, T = RR_{1}  generate  12  D_{¥}  
P = RR_{1}R_{2}  generates  1g  C_{¥}  
X  generates  11  C_{¥}  
The survey of the characteristics of the symmetry groups of friezes relies on the work of A.V. Shubnikov, V.A. Koptsik, 1974; H.S.M. Coxeter, W.O.J. Moser, 1980.
The first derivation of the symmetry groups of friezes as the line subgroups of the symmetry groups of ornaments G_{2} and their complete list, was given by G. Pòlya (1924), P. Niggli (1926) and A. Speiser (1927).
Cayley diagrams (Figure 2.28):