This is a presentation for the dihedral group , better known as the Klein group, the non cyclic
group of order 4
A dihedral group can be generated by two distinct elements of order 2
(a) Show that this definition allows only one infinite group up to isomorphism and determine its centre.
As already noted, the above is not an infinite group but a pretty finite one...
Is this right:
The only non-trivial rotation that commutes with all reflections is the rotation over . So is a rotation by and is a reflection.
Since . Therefore,
So , which is a contradiction; thus, no element of the form is in the center.
Similarly, if for , then , which is possible only if . Hence, commutes with iff
So if , the the center of is ; if is odd the center is .