Let G be the symmetric group Sym({1,2,..., 20}). Prove that G has a subgroup H such...

 

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Let G be the symmetric group Sym({1,2,..., 20}). Prove that G has a subgroup H such that H≈ A5 and such that for any distinct x, y € {1,2,..., 20} there exists h E H such that h(x) = y. Hint: Consider the rotational symmetries of the regular dodecahedron shown below. (2,5) (1,4) (4,3) (3,1) (3,2) (4,2) (5,3) (5,1) (1,2) (4,5) (5,4) (2,1) (1,5) (2,4) (3,5) (2,3) (1,3) (3,4) (4,1) (5,2) Let G be the symmetric group Sym({1,2,..., 20}). Prove that G has a subgroup H such that H≈ A5 and such that for any distinct x, y € {1,2,..., 20} there exists h E H such that h(x) = y. Hint: Consider the rotational symmetries of the regular dodecahedron shown below. (2,5) (1,4) (4,3) (3,1) (3,2) (4,2) (5,3) (5,1) (1,2) (4,5) (5,4) (2,1) (1,5) (2,4) (3,5) (2,3) (1,3) (3,4) (4,1) (5,2)