Can someone help me with this proof? It was an exam problem, but I can't get it.
D_2n = < r, s | r^n = s^2 = 1, rs = sr^(-1) >
If n = 2k is even and n>= 4, show that z = r^k is an element of order 2 which commutes with all elements of D_2n. Show also that z is the only nonidentity element of D_2n which commutes with all elemtns of D_2n.
Obviously, the identity element commutes with all elements in . But I am looking for a nonidentity element, so from the first proof, is the nonidentity element that commutes with all elements in . But how do I know that it is the ONLY nonidentity that commutes with those elements?
So I let where , then and . So commutes with all elements of no matter what a and b are, isn't it? Then the nonidentity element in the form of also commutes with all elements of , so is not the only nonidentity element that commutes with the elements in ...?
Am I thinking this right?...
Let us consider two cases: (i) , (ii) , .
In case (i) we have that commutes with . It definitely commutes with no matter what is but to commute with we need to have and so this forces (where is such an integer so ). Thus, we find that commutes with all the elements.
You try case (ii).