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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 , .

Answer:

.

For ,

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.

Therefore,

For ,

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