Decompose the product set u x v into orbits

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Reference no: EM13178758

1. Let l_1 and l_2 be lines through the origin in R^2 that intersect in an angle pi/n, and let r_i be the reflection about l_i. Prove that r_1 and r_2 generate a dihedral group Dn.

2. The symmetric group S_3 operates on two sets U and V of order 3. Decompose the product set U x V into orbits for the "diagonal action" g(u, v) = (gu, gv), when

A) the operations on U and V are transitive

B) the operation on U is transitive, the orbits for the operation on V are {v_1} and {v_2, v_3}

3. Let G be the group of symmetries of a cube, including the orientation-reversing symmetries. Describe the elements of G geometrically.

Reference no: EM13178758

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