Point group

Definition

Symbol-free definition

A point group in dimension $n$ is a subgroup of the orthogonal group $O_n(\R)$, or equivalently, a group of geometric symmetries which leave a particular point fixed.

When we use the term point group, we usually mean not just the abstract group, but rather its embedding as a subgroup of the orthogonal group.

A point group can thus also be regarded as a faithful linear representation of a group in terms of only isometries.

Related notions

• Rosette group: A point group in two dimensions
• Molecular point group: A point group in three dimensions, so named because these groups describe the symmetries of molecules
• Space group
• Crystallographic point group