What is the symmetry of this group?

Question

My lecturer tends to describe groups as representing the symmetry of some shape, and I've read in various places that groups can be defined based on objects that are invariant under a transformation they represent. So I presume O(2) would be two circles, a rotated and reflected one, and SO(2) is a single circle. So if I have a set of pairs (A, v) which form a group, where A is part of SO(2) and v is a real 2D vector, what symmetry does this represent? I can't see what shape a 2-component column vector would be, although I understand why SO(2) is a circle. And certainly I don't know what symmetry their combination might represent.

The operation of this group, symbol $-$, hasn't really been defined, beyond $$(A, u) - (B, v) = (AB, Av + u) $$ and it clearly doesn't represent subtraction.

Thanks for any help, or resources on the subject!

Answer 1

The group law you have mentioned comes from the affine group $Aff(V)$, consisting of pairs $(A,v)$, with a matrix $A$ and a vector $v$, with group law $$ (A,v)\circ (B,w)=(AB,v+Aw). $$ This is explained in detail here. The second part of your question is difficult to answer, because we need to know the precise definitions. Plane Symmetry groups is a good key word to look for. The isometry group $Iso(\mathbb{R}^2)$ is a subgroup of the affine group $Aff(\mathbb{R}^2)$. Then $A,B$ are in $O(2)$.

Answer 2

In continuity with the good answer of Dietrich Burde, taking $A \in O(2)$, gives a subgroup of the Affine group of the plane, i.e., the group of motions in the plane (combinations of rotations, symmetries and translations).