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2 Exercises 1. For a square let’s consider D4 . There are four rotational symmetries: e, r1 , r2 , r3 where π rn denoting rotation clockwise by 2 n radians, h a reflection across a horizontal line v a reflection across a vertical line, u an upward diagonal axis of reflection, d a downward diagonal axis of reflection. Complete the following group table: ∗ e e e r1 r1 r2 r2 r3 r3 h v u d r1 r1 r2 r3 e r2 r2 r3 e r1 r3 h v u d r3 h v u d e r1 r2 2. Prove that Dn is nonabelian for n ≥ 3 . 3. Prove that Dn is noncyclic for n ≥ 3 .

Square red, square . . blue square, red . . blue What is going on here? It’s hard to tell exactly. We can see that for pair of elements from the set {circle, blue, square, red} a third is mentioned. It’s unclear what connection the third element has with the first two, but it appears that some rule dictates what happens. What does this situation have to do with abstract algebra? It’s actually one that you have encountered before. com 36 Click on the ad to read more An Introduction to Abstract Algebra Group Theory 8, 7 · · · 15 2, 3 · · · 5 12, 2 · · · 14 It’s easy to see whats going on in this situation.

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