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Mathematics 466 A. Hulpke Homework (due Dec. 5) 56) If G and H are groups, the cartesian product G × H becomes (as in problem 33 for rings) a group with component-wise operations: (a, b) ⋅ (c, d) ∶= (ac, bd), (a, b)−1 ∶= (a −1 , b−1 ) It is denoted the direct product G × H or sometimes (for abelian groups) the direct sum G ⊕ H. a) Show that (for finite groups) G × H is cyclic if and only if G and H are cyclic, and gcd(∣G∣ , ∣H∣) = 1. Hint: Show that if G × H is cyclic with generator (a, b), then a must be a generator for G, and b for H. Vice versa, consider the order of (a, b). b) Show that If a = b ⋅ c with gcd(b, c) = 1 then Za ≅ Zb × Zc . 57∗ ) (Considering the permutations of the top cubies of the 2 × 2 × 2 cube) Show that S4 = ⟨(1, 2, 4, 3), (1, 2)⟩. 58) Let G be the group of rotational symmetries of a cube. Show that G ≅ S4 . (Show that the action on the diagonals is faithful.) 59) A dodecahedron has (as calculated in problem 55) 60 A symmetries, namely C 1 identity B A 6 ⋅ 4 = 24 rotations through face centers, B 2 ⋅ 10 = 20 rotations around corners, C 15 rotations around edge centers. a) For elements of type A,B and C, determine the orbits (of the cyclic subgroup generated by this element) on the faces. b) Determine how many different ways (up to symmetry) exist, to color the faces of the dodecahedron in green and gold. B 60) Let G be a group. We have seen that G acts on G via the action µ(x, g) ∶= g −1 x g. The orbits under this action are called the conjugacy classes of G. a) Show that every normal subgroup of G is a union of conjugacy classes. b) Determine the conjugacy classes of S5 and their cardinalities. (You can use the GAP command cl:=ConjugacyClasses(SymmetricGroup(5)); List(cl,Size); to verify the combinatorial counts of the cardinalities.) c) Show that S5 can have normal subgroups only of orders 1, 60, 120. Problems marked with a ∗ are bonus problems for extra credit.