ahw4 - Math 3124 Thursday, September 22 Fourth Homework...

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Math 3124 Thursday, September 22 Fourth Homework Solutions 1. Problem 8.3 on page 50 Determine the group of symmetries of an equilateral trian- gle. ± ± ± ± ± ± ± ± ± ± ± ±A A A A A A A A A A A A B A C Let the equilateral triangle be ABC as shown. Any symmetry must permute the ver- tices, and any symmetry which fixes all the vertices must be the identity. Therefore there are at most 6 symmetries, and we shall see next that there are exactly 6. In fact we have the identity which we denote by (A), reflection in the line through A to the midpoint of BC which we shall denote (BC), reflection in the line through B to the midpoint of CA which we shall denote (AC), reflection in the line through C to the midpoint of AB which we shall denote (AB), rotation anticlockwise through 120 which we shall denote (ABC), and rotation through 240 which we shall denote (ACB). Now the composition of mapping is obtained by composing the corresponding permutations. Therefore the Cayley table is (A) (BC) (CA) (AB) (ABC) (ACB)
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ahw4 - Math 3124 Thursday, September 22 Fourth Homework...

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