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Homework Nine Part I (of II) (1) Let S be a finite set with n elements. How many different permutations of S are there? Prove your answer. (2) Describe a group structure on the set of moves of a Rubik’s cube. If you are unsure what these moves are, you might find the first few seconds of this video helpful. Your description should include your chosen identity element, chosen binary operation and a brief explanation for why the group axioms hold. (3) Let G be a finite group (i.e. a group with finitely many elements) with binary operation ? and identity eG . Prove that for every g ∈ G there is some n ∈ N such that (†) g ? g ? g ? . . . ? g = eG . n times (4) (5) (6) (7) It might be helpful to figure this out concretely for your favourite permutation group before tackling the case of an abstract group. What does (†) mean in the context of the Rubik’s cube? Let Γ and Ξ be graphs and suppose that there is a surjective graph homomorphism ϕ : Γ → Ξ. Prove that if Γ is connected then Ξ is also connected. Prove or disprove: there is a graph homomorphism from a square graph to a graph consisting of a single edge (the answer is not what you initially think it is). Prove that a graph whose vertices and edges can be drawn as such is not isomorphic to a graph whose vertices and (here, as in the previous edges can be drawn as such question, it might be helpful to label the vertices before trying to write anything down). Find all the automorphisms of your favourite graph. In other words, describe its automorphism group. 1