Math 3230 (Roby) Review & Practice Final Solutions Fall 2014 SHOW ALL YOUR WORK! Make sure you give reasons to support your answers. If you have any questions, do not hesitate to ask! No calculators are to be used, but you may bring one 8.500 × 1100 sheet (two-sided) of notes to class with anything you like written on it. The usual Cayley tables are provided at the end of this exam. The final will be comprehensive, following the outline below. The problems given are just a few examples, focussing more on the later material in the course (since we already have a practice midterm). They do not cover the full range of problems that could be asked on an exam, but give a good idea when combined with the problems from homework, quizzes, midterm and the practice midterm. Here are some specific tasks I expect you to be able to perform with demonstrated understanding. In all of the following, G denotes a group. 1. Decide whether a given set and operation forms a group or not. 2. Given the Cayley (group operation) table for a group G, perform computations in G and determine whether certain properties hold in G. 3. Be able to prove basic properties of groups directly from the axioms. 4. Decide whether a given subset of a group is actually a subgroup. 5. Perform computations in and know basic properties of specific examples of groups including: Zn , U (n), Sn , Dn , GL(2, F), SL(2, F), Z, Q+ , R∗ , C∗ , Rn . 6. Disinguish between properties that hold only for finite groups (vs. infinite ones), only for abelian groups (vs. non-abelian ones), and only for cyclic groups. 7. Know and be able to use the Fundamental Theorem of Cyclic Groups and other basic facts about cyclic groups, e.g., how to find the number of elements of order d in a cyclic group. 8. Understand how to decompose a permutation into disjoint cycles and as a product of transpositions and how to multiply permutations given as products of cycles. Determine easily whether a permutation is odd or even. 9. Understand what the order of an group and the order of an element are, and how these are related to one another in a group. Be able to compute the order of elements in various groups. 10. Know the definitions of the center of a group, Z(G), and the centralizer of a group element, C(a). 11. Know the basic properties of isomorphisms of groups. Understand that the set of isomorphisms of a group is itself a group, namely Aut(G), which contains the subgroup Inn(G). 12. Prove that two groups are isomorphic (by giving an explicit map and checking that it’s an isomorphism) or that they are not isomorphic (by giving examples of a property that only one of the two groups has). 13. For H ≤ G, be able to define cosets of H in G and prove their basic properties, particularly that they partition G into disjoint subsets. 14. Know the definitions of basic properties of both external and internal direct products of groups; understand how cyclic groups and U -groups behave with respect to direct products. 15. Understand the central importance normal subgroups as being those whose cosets form quotient (aka factor ) groups. Be able to construct examples, and understand how properties of groups relate to properties of their quotient groups. 16. Know the definition and basic properties of homomorphisms. Understand that their kernels are always normal subgroups (and vice-versa). 17. Know how to carefully state the following theorems and apply them in different situations: (a) Cayley’s Theorem (and proof!); (b) Lagrange’s Theorem (and proof!); (c) Orbit-Stabilizer Theorem; (d) The G/Z Theorem; (e) G/Z(G) = Inn(G); (f) First Isomorphism Theorem; (g) Cauchy’s Theorem for Abelian Groups; (h) Classification of all groups of order 2p and of order p2 , where p is a prime; (i) Fundamental Theorem of Finite Abelian Groups; (j) The Class Equation and how it shows that p-groups have nontrivial centers (and proofs!); and (k) The Sylow Theorems; 18. Understand the theory of the course well enough to distinguish true statements from false ones, giving proofs or counterexamples as appropriate. Here are a few sample problems (not meant to be comprehensive or a template for the final itself). 1. Decide whether each statement below is Always, Sometimes, or Never True. Justify your answer. (a) If G is a finite abelian group with #G divisible by 12, then G has a cyclic subgroup of order 4. Sometimes true. Z12 has the cyclic subgroup h3i of order four, but Z2 ⊕ Z2 ⊕ Z3 has no subgroup (not even an element) of order four. (b) A group of order 221 is cyclic. Always true, since 221 = 13 · 17, using Thm. 24.6. (c) If #G = p` , where p is a positive prime and ` ≥ 1, then Z(G) is trivial. Never, by Thm. 24.2. 2. Let H be a subgroup of a group G. (a) Prove that the number of conjugates of H in G is |G : N (H)|. We want to show that the map τ : G/N (H) → {conjugates of H} given by τ (xN (H)) = xHx−1 is a well-defined bijection. (We don’t need to worry about operation-preserving here.) Note that xN (H) = yN (H) ⇐⇒ y −1 xN (H) = N (H) ⇐⇒ y −1 x ∈ N (H) ⇐⇒ y −1 xH(y −1 x)−1 = H ⇐⇒ xHx−1 = yHy −1 , showing τ is well-defined and one-to-one. Since, τ is clearly onto, we are done. (b) If H is a proper subgroup of a finite group G, show that G is not the union of all conjugates of H. Since each conjugate of H has the same order as H and contains the identity, we know by the previous part that the union of all conjugates of H has fewer elements than |H| · |G : N (H)|. But since H ⊆ N (H), this is smaller |G| = |G|. than |H| · |H| 3. For each pair of groups below, decide whether they are isomorphic or not and prove your answer is correct. (a) The group of rotations of a cube (in three dimensions) and Stab1 S5 . They are both isomorphic to S4 by Thm. 7.5 and an easy isomorphism. (b) Z60 and Z2 ⊕ Z2 ⊕ Z3 ⊕ Z5 . LHS ≈ Z4 ⊕ Z3 ⊕ Z5 and has elements of order 4, which don’t exist in RHS. (c) Z2 ⊕ Z4 ⊕ Z10 ⊕ Z45 and Z2 ⊕ Z10 ⊕ Z180 . Both are isomorphic to Z2 ⊕ Z2 ⊕ Z4 ⊕ Z5 ⊕ Z5 ⊕ Z9 . (d) Aut(Z9 ⊕ Z7 ) and Z6 ⊕ Z6 . LHS ≈ Aut(Z63 ) ≈ U (63) ≈ U (9) ⊕ U (7) ≈ Z6 ⊕ Z6 . 4. Compute each of the following: (a) The number of elements of order 6 in S6 ? If σ ∈ S6 has order 6, then it must have one of the cycle types: (6) or (3, 2, 1). There are 6!/6 = 120 of the first type and 63 · 2! · 32 = 120 of the second type, for a total of 240. (b) In any group of order 80, the maximum number of elements of order 5 and the minimum number. Can you give examples of specific groups achieving these extreme values? By third Sylow, n5 ≡ 1 (mod 5) and n5 | 16, forcing n5 = 1 or 16. In the former case, we have one (cyclic) subgroup of order 5 with 4 elements of order 5, for example Z80 . In the latter, we have 16 subgroups of order 5, each containing the identity, for a total of 16 · 4 = 64 elements of order 5. Such a group would have to be non-Abelian (why?) and is harder to describe explicitly. (c) The smallest possible odd integer that can be the order of a non-Abelian group. For any primes p < q where p - q − 1, groups of order p and pq are cyclic (Cor. 3 of Thm. 7.1 and Thm. 24.6); groups of order p2 are Abelian (Cor. to Thm. 24.2). The smallest odd integer not of this form is 21. Such a group can be constructed explicitly using the Sylow theorems (as an easy websearch reveals). 5. Show that there is no homomorphism from G = Z8 ⊕ Z2 ⊕ Z2 onto Z4 ⊕ Z4 . If there were such a homomorphism, then it would induce an isomorphism from G/(Ker ϕ) to Z4 ⊕ Z4 . Note that ϕ(4, 0, 0) = 4ϕ(1, 0, 0) = 0, so Ker ϕ = {(0, 0, 0), (4, 0, 0)}. Now check that G/ Ker ϕ has seven elements of order 2, while Z4 ⊕ Z4 has only three. 6. Let H E S4 with #H = 4. Prove that S4 /H ≈ S3 . Since |S4 /H| = 6 = 2 · 3, by Thm. 7.3 S4 /H is isomorphic to Z6 or D3 ≈ S3 . We can rule out the former because S4 has no elements of order 6. (Why?) 7. Find all groups G of order 72 · 112 . Let H be a 7-Sylow subgroup and K a 11-Sylow subgroup of G. Then by Sylow’s third theorem, n7 = n5 = 1. Thus, H and K are the unique Sylow subgroups of their respective orders, hence normal in G. Now H ∩ K = {e} (why?), and we claim that elements of H commute with elements of K. If h ∈ H and k ∈ K, then hkh−1 k −1 = (hkh−1 )k −1 ∈ Kk −1 = K and hkh−1 k −1 = h(kh−1 k −1 ) ∈ hH = H. Hence, hkh−1 k −1 ∈ H ∩ K =⇒ hkh−1 k −1 = e =⇒ hk = kh. Therefore, G = H × K is the internal direct product of abelian groups and must be abelian. By FTFAG, we have four possibilities: Z772 , Z25 ⊕ Z7 ⊕ Z7 , Z5 ⊕ Z5 ⊕ Z49 , and Z5 ⊕ Z5 ⊕ Z7 ⊕ Z7 . 8. Go back over your old homework, quizzes, midterm, and the practice midterm to review and make sure you understand any problem on which you lost points. Check!