Is there a difference between the isometry group and the symmetry group? Obviously the first one implies the existence of a metric but if we have a simple pentagon, are both groups isomorphic to $D_5$ or do we need to differ here?
Asked
Active
Viewed 427 times
2
-
In the example of a cube, the rotations are the full Symmetric group $S_4$, though the symmetries of a cube form a group of size $48$. The isometries are reflections that can't be physically realized. – J. Linne Sep 04 '24 at 15:34
1 Answers
1
It depends on the definition of symmetry group. In the context of ornament groups and crystallographic groups we certainly have have the following definition. For a Euclidean vector space $E$, a symmetry of a set $M$ in $E$ is an isometry $s\colon E\rightarrow E$ satisfying $s(M)=M$. It follows that all symmetries of $M$ form a subgroup $\operatorname{Sym}(M)$ of the isometry group $\operatorname{Isom}(E)$.
Dietrich Burde
- 140,055