7. The center of D4 is defined by Z(D4) = {0 E D4|0oT = Too for all T E D4}. For

example, we have & E Z(D4) because & O T = TO ε for all T E D4.

Note: The notation Z(D4) originates from the German word Zentrum ("center").

(a) Describe the set Z(D4) in your own words. How does the center differ from a

centralizer?

(b) Find all the elements of Z(D4).

WEW

(c) Recall that C(180) = {0 E D4 | 0 0 r180 = 180° 0}. Find all elements of C(r180).

(d) Elizabeth says, "After doing part (b) of this problem, I could do part (c) without

looking at the composition table for D4." Why might she say that?

(This exercise is referenced in Example 11.15.)