Show, by means of an example, that the group of symmetries of a subset X of a Euclidean space is, in general, smaller than Sym(x).

Question

Show, by means of an example, that the group of symmetries of a subset X of a Euclidean space is, in general, smaller than Sym(x).

Been slightly stuck on this one for a while - I'm not too sure on the exact difference between the group of symmetries and Sym(x) either. Anyone able to help?

Answer 1

By definition ${\rm Sym}(X)$ is the set of all bijections $f\colon X\rightarrow X$. This set together with composition forms a group. If $|X|=n$, then ${\rm Sym}(X)=S_n$ is the symmetric group with $n!$ elements. However, the group of symmetries, say , of a regular $n$-gon is the dihedral group $D_n$ of $2n$ elements. Clearly, $2n<n!$ for all $n\ge 4$.