We often think of symmetry as a property of shapes. Another way of thinking about it is as something you do to an object which keeps the object looking the same. The example you’ve likely met is reflection. The other one that we’ll consider today is rotation. An important feature is that we consider ‘doing nothing’ as a symmetry - we call this the identity.
Note the two following features of symmetries
Applying a symmetry and then a second symmetry gives us another symmetry (which may be the same as one of the two we’ve used!)
Each symmetry has an inverse. Suppose we apply symmetry \(x\). Then there is some symmetry we can apply after \(x\), which means that overall, we’ve applied the identity.