Math 113 Midterm ExamJuly 16th, 2009NameQuestion Score Possible1.1 21.2 31.3 41.4 32.1 32.2 52.3 4P2411 ComputationsProblem 1.1. (2 points) Let σ ∈ S7be the element σ = (1235)(237)(45).Write σ as a product of disjoint cycles, and compute the order of σ.Problem 1.2. (3 points) Find all possible cycle types of permutations inS6. For each cycle type, state whether permutations of that cycle type areeven or odd. Circle the cycle types which correspond to elements of A6.2Problem 1.3. Let G = (Z/11Z)×. Do the following:1. (1 point) Show that G is cyclic by finding a generator.2. (3 points) List all subgroups of G, and for each subgroup of G, give agenerator for that subgroup.3Problem 1.4. (3 points) Show that Z/6Z∼=Z/2Z × Z/3Z by writing downan explicit isomorphism f : Z/6Z → Z/2Z × Z/3Z. (You must prove thatyour function f is in fact an isomorphism).42 TheoryProblem 2.1. (3 points) Show that Dnhas a subgroup of order k for everyk dividing n.5Problem 2.2. Let G be an abelian group, and let g, h ∈ G be elements.1. (2 points) Let H be the set {gnhm| n, m ∈ Z}. Show that H is asubgroup of G.2. (3 points) Suppose that there exists some element a ∈ G and integersk, ` ∈ Z such that g = akand h = a`. Show that the subgroup Hdefined in part (1) is cyclic.6Problem 2.3. (4 points) State and prove Lagrange’s