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hw1sol - Homework 1 Solutions 1 There are six symmetries of...

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Homework 1 Solutions 1. There are six symmetries of an equilateral triangle: the identity, rotate clockwise by 120 rotate counterclockwise by 120 , and three mirror symmetries about lines which pass through one vertex and the opposite edge. In class we created a partial multiplication table for these symmetries. Complete this table. Solution: Define 1 to be the identity transform, ρ to be rotation by 120 (so ρ 2 is rotation by 240 . Let σ 1 be the reflection through the vertical line (through the center), σ 2 to be the reflection through the line at 30 , and σ 3 to be reflection through the line at - 30 . Then depending on the interpretation of order, our Cayley table is one of the following: 1 ρ ρ 2 σ 1 σ 2 σ 3 1 1 ρ ρ 2 σ 1 σ 2 σ 3 ρ ρ ρ 2 1 σ 3 σ 1 σ 2 ρ 2 ρ 2 1 ρ σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 1 ρ ρ 2 σ 2 σ 2 σ 3 σ 1 ρ 2 1 ρ σ 3 σ 3 σ 1 σ 2 ρ ρ 2 1 1 ρ ρ 2 σ 1 σ 2 σ 3 1 1 ρ ρ 2 σ 1 σ 2 σ 3 ρ ρ ρ 2 1 σ 2 σ 3 σ 1 ρ 2 ρ 2 1 ρ σ 3 σ 1 σ 2 σ 1 σ 1 σ 3 σ 2 1 ρ 2 ρ σ 2 σ 2 σ 1 σ 3 ρ 1 ρ 2 σ 3 σ 3 σ 2 σ 1 ρ 2 ρ 1 2. Construct a group G with | G | = 3 (so G has exactly three elements).
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