symmetry group

symmetry group (plural symmetry groups)

1. (geometry, algebra, group theory) A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.
   • 1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, published 2000, page 76:

     [The] situation is more delicate in the case of higher dimensional symmetry groups; it is not in general possible to reduce the order of an equation invariant under an r-parameter symmetry group by r using only quadratures.

   • 2000, C. E. Horne, Geometric Symmetry in Patterns and Tilings, page 79:

     It is often assumed that the design unit inside each fundamental region (particularly for monotranslational symmetry groups p111 and p1a1 and ditranslational symmetry groups p1 and pg) is asymmetric.

   • 2010, Stanislav Jendrol, František Kardoš, “28: Symmetry of Fulleroids”, in Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 28–1:

     The structure of the symmetry group of a molecule affects several spectroscopic aspects and vice versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.
     In this chapter, we study symmetry groups of fulleroids.

Not to be confused with symmetric group.

• (group of transformations): full symmetry group (emphasises the inclusion of orientation-reversing transformations)

• (group of transformations): lattice, point group, rotation group, space group

• (group of transformations): discrete symmetry group, proper symmetry group

• isometry group
• Lie group