- #1

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## Homework Statement

If ##A = \{1,r^2,s,sr^2 \}##, compute ##C_{D_8}(A)##

## Homework Equations

## The Attempt at a Solution

So I am not exactly sure what is the best way to do this, but here is what I tried. From previous problems I know that ##Z(D_8) = \{1, r^2 \}##. So ##Z(G) \le C_{D_8}(A) \le D_8##, so the order of ##C_{D_8}(A)## can only be 2, 4, or 8. It's not 8 since we can compute directly that ##r \not \in C_{D_8}(A)##. It's not 2 since we can compute directly that ##s \in C_{D_8}(A)##. So the order is 4. And since by computation we see that ##sr^2 \in C_{D_8}(A)##, we have that ##C_{D_8}(A) = A##.