I have:

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.

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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 .