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Question: What is , the center of the dihedral group ?
Algebraically, the dihedral group may be viewed as a group with two generators and , i.e. with , .
Proof: For , which is abelian. Thus, .
For , , the Klein four-group, which is also abelian. Thus, .
Let , . Clearly elements in commute with each other.
Let be an element in . (). Let be an element in . ()
I.e. the only element in (other than 1) that is in the center is , which is only possible if is even.
Let , be two distinct elements in . ()
By earlier analysis, this is true iff . Each is not in the center since we may consider , i.e. . Then . (since ). also does not commute with for the same reason.