denotes a subset of Rn
then its symmetry group
is the group of isometries
that leave the subset X
stable. For examples of isometries
see the write-ups on isometries of the plane
and the orthogonal group
It may be that every symmetry of X is forced to fix the origin
and so be a linear transformation. This is the case for a regular polygon
centred at the origin in R2, for example, and for
also for a Platonic solid centred at the origin in 3-space.
In that case Symm(X) will be a subgroup of the orthogonal group
O(n) and we can also consider its direct (or rotational) symmetry
group Rot(X) which is the intersection of Symm(X)
with the special orthogonal group SO(n).
We are mostly interested in the case of n=2 and n=3. In that
case Rot(X) consists of rotations, as its name suggests.