# 3. Why study groups?#

See VGT Â§ Errata for known issues in this chapter.

## 3.1 Groups of symmetries#

This section discusses Symmetry group. See also Symmetry (disambiguation), in particular Symmetry in mathematics:

Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.

When the author says â€śsimilarâ€ť in Definition 3.1 he means it in a technical sense; see Similarity (geometry). Youâ€™ll often see â€śLâ€ť used to make this check easier, as in Dihedral group of order 6.

In this section, the invariant we are pursuing is the occupation of ambient space. Some things change and some stay the same; itâ€™s not like rotating a molecule means â€śnothing changedâ€ť but that nothing changed regarding the metrics we care about. What does change in this context is e.g. the labeling we apply to parts of the object, which are considered not part of the ambient space. If we were to write them on the objects with marker, we would imagine the marker as taking no space.

### 3.1.1 Shapes of molecules (âš )#

See the VGT Errata for Figure 3.5.

### 3.1.3 Art and Architecture#

See also Glide reflection. The author specifically mentions the seven frieze groups of Frieze group Â§ Descriptions of the seven frieze groups and the seventeen wallpaper groups of Wallpaper group Â§ The seventeen groups.

## 3.2 Groups of actions#

The misspelling of â€śDefintionâ€ť is not mentioned in the errata.