I hear all the time people saying things like "groups are the algebraic equivalent to the notion of symmetry". Yet I don't think I've ever really understood what was meant by this. I'm familiar with groups of symmetries (like the dihedral groups), but in these cases, to me it just seems like we are using the language of groups to talk about the ordinary notion of symmetry. It seems like people are saying all groups have "something to do with an algebraic notion of symmetry" What this might be though is unclear to me. For example, what does the group GL_2(R) have to do with symmetry?