Math 120. Basic Algebra Midterm - Practice Questions
Note: These problems are only intended to help you study. Questions based an any material covered in class may appear on the actual test.
1. True or false? (a) The set of all positive real numbers is a group under multiplication? (b) For all a, b ∈ G, where G is a group, (ab)3 = a3 b3 . (c) If α, β are disjoint cycles in S7 then αβ = βα. (d) Every transposition is an even permutation. (e) Every subgroup of Sn has order dividing n!. (f) Let G be a group and H ≤ G, K ≤ G cyclic subgroups. Then the intersection H ∩ K is a cyclic subgroup. (g) Any two groups of order 17 are isomorphic. (h) Any two groups of order 24 are isomorphic.
2. Let Q∗ be the set of strictly positive rational numbers. Show that Q∗ forms a group with the binary + + law given by multiplication. Is this group cyclic? 3. Find the signature sgn(α) and α−1 , where: α = 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 .
4. What is th order of each of the following permutations? (a) (b) 1 2 1 7 2 1 2 6 3 5 3 1 4 4 4 2 5 6 5 3 6 3 6 4 7 5 .
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5. Let: α = 1 2 2 1 3 3 4 5 5 4 6 6 and β = 1 6 2 1 3 2 4 4 5 3 6 5 .
Compute each of the following: α−1 , βα, α2 β. 6. Suppose that G is a finite commutative group and that G has no element of order two. Show that the map Θ : G → G, Θ(x) = x2 is a group isomorphism of G to itself. Will this statement still hold if G is infinite? 7. Let G be a group. Prove that the function α : G → G, α(g) = g −1 is a group isomorphism if and only if G is commutative. 8. Let G be a group for which x2 = e for any x ∈ G. Prove that G must be commutative. 9. Let G be a finite group. Show that the number of elements x ∈ G such that x3 = e is odd. Show that the number of elements x ∈ G such that x3 = e is even. 10. For each of the groups in the following list, describe all the proper subgroups. Z/8Z, Z/12Z, D4 , Z.
11. Let G be a group and x ∈ G. If x2 = e and x6 = e, what is the order of x? 12. List all the elements of Z/40Z that have order 10. 13. Let G be the set of all polynomials of the form ax2 + bx + c, with coefficients from the set {0, 1, 2}. One can make G a group under addition, by adding polynomials in the usual way, except that one uses remainders modulo 3 to combine the coefficients. What is the order of this group? Is G cyclic? 14. Prove that a commutative group of order 6 must be cyclic. 15. Let H ≤ S3 , given by H = {e, (12)}. Is H a normal subgroup of S3 ? 16. Prove that An (the set of even permutations) is a normal subgroup of Sn .
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