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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?

Buh
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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
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