MATH 113 HOMEWORK 3DUE MONDAY, JULY 13TH1. Basic ComputationsProofs can be omitted in this section. You should still give enough expla-nation that your classmates could learn the technique from your calculations.Problem 1.1. Do Judson Ch. 3 Exercises 20 and 24.Problem 1.2. Consider the permutationsσ =1 2 3 4 5 6 76 7 4 3 1 5 2, τ =1 2 3 4 5 6 71 3 4 5 7 6 2in S7. Do the following:(1) Decompose σ and τ into cycles.(2) Compute στ and τσ.(3) Compute the order of σ, τ, στ , and τσ.(4) Determine the signs of σ, τ, στ , and τσ.Problem 1.3. Find all possible cycle structures in S7. Explain which ofthese cycle structures correspond to elements of A7. Use this to calculateall possible orders of elements of A7.Problem 1.4. Do Judson Ch. 5 Exercise 5.2. Basic ProblemsProblem 2.1. Let G be a group and let a, b ∈ G be arbitrary. Show that:(1) The order of a is equal to the order of b−1ab.(2) The order of ab is equal to the order of ba.Problem 2.2. Let G be a group and let H and K be subgroups of G.(1) Show that H ∩ K is a subgroup of G. Conclude that H ∩ K is alsoa subgroup of H and a subgroup of K.(2) Suppose that H is cyclic of order 15 and K is cyclic of order 16.What is the order of H ∩ K?Problem 2.3. Let G be a group.(1) Show that if G is abelian, then the set of elements of finite order inG form a subgroup of G. This is called the torsion subgroup of G.(2) Give an example that shows that if G is not assumed to be abelian,then the set of elements of finite order in G can fail to be a subgroupof G.1MATH 113 HOMEWORK 3 DUE MONDAY, JULY 13TH 2Problem 2.4. (Judson Ch. 4 Exercise 6) Find all of the subgroups ofA4. Calculate the order of each subgroup you find. Notice that there is nosubgroup of order 6. Explain why you might find this surprising.Problem 2.5. (Judson Ch. 4 Exercise 18) Show that Anis nonabelian forn ≥ 4.Problem 2.6. (Judson Ch. 4 Exercise 26) Show that any permutation inSncan be written as a product of the transpositions (12), (13), . . . , (1n).Show likewise that it can be written as a product of the transpositions(12), (23), . . . , ((n − 1) n). Finally, show that it can be written as a productof the two permutations (12) and (12 . . . n).3. Creative ProblemsProblem 3.1. (Other groups of symmetries) We’ve now seen groups of“symmetries of a set” (the symmetric group Sn), “symmetries of the n-gon” (the dihedral group Dn), and a few other groups of symmetries (forexample, the symmetries of a rectangle don’t fall into either class above).Now consider the following:(1) The “group of symmetries of the integers”: Let G be the set offunctions f : R → R of the form f(x) = ax + b, where a, b ∈ R, suchthat f induces a bijection from Z to Z (for example, f (x) = x + 1is such a function, but f (x) = 2x + 1 is not). Describe all possiblea and b. Show that G is a group under composition. Try to find asmany interesting properties of G as you can.(2) Now come up with TWO other interesting examples of “groups aris-ing from symmetries”; you might look at symmetries of your favoriteshape, or symmetries of the tile pattern on your bathroom wall. Tryto say as much as you can about the groups you discover.Problem 3.2. Let G be a group, let g be an element of G, and let H be asubgroup of G. Do the following:(1) Show that gH = Hg if and only if gHg−1= H; here gHg−1={ghg−1| h ∈ H}.(2) Part (1) shows that multiplying subgroups of G by elements of G“behaves like multiplying elements of G” in some sense. Try to findother examples of this. Try to find cases where this fails (for example,what happens if you replace g here by another subgroup K of G?).See how general of a statement you can