# Homework ## Homework 12, due Apr 24 at 12:00pm Please do the following problems from the textbook: Isaacs p96: 7.5a, 7.10 and the following additional problems **A.** Describe *A*_{4} as a semidirect product. Make sure that you describe the action by automorphisms! **B.** Explain why the quaternion group *Q* cannot be decomposed as a semi-direct product. Hint: examine the subgroup lattice! ## Homework 11, due Apr 19 at 12:00pm Please do the following problems from the textbook: Isaacs p81: 6.8, 6.12 Isaacs p96: 7.1, 7.2, the *n*=2 case of 7.3 and the following additional problems **A.** Draw the subgroup lattice of **Z**_{5} x *S*_{3}. Indicate normal subgroups with diamonds and non-normal subgroups with ovals. **B.** Draw the subgroup lattice of **Z**_{3} x *S*_{3}. Indicate normal subgroups with diamonds and non-normal subgroups with ovals. Your answers to A and B should be slightly different! ## Homework 10, due Apr 12 at 12:00pm Please do the following problems from the textbook: Isaacs p81: 6.4, 6.5, 6.7, 6.13 and the following additional problems **A.** Let *G* = *SL*_{2}( **Z**_{7} ) / ( *Z*( *GL*_{2}( **Z**_{7}) \cap *SL*_{2}( **Z**_{7} ) ), where \cap denotes intersection. i. Verify that |*G*| = 168. ii. Exhibit at least 2 subgroups of *G* with order 3, and at least 2 subgroups of *G* with order 7. (It may be helpful to first find these subgroups in *SL*_{2}, and show that they are preserved in the quotient.) **B.** Let *G* be as in the previous problem. By following the proof technique used to show that *A*_{5} is simple, show that *G* is a simple group if *G* has a proper normal subgroup, then it has a normal 2-subgroup. You may use without proof the fact that *G* does not ~~embed injectively into ~~*S*_{7} have 7 Sylow *p*-subgroups.
**C.** Describe the isomorphism type of a Sylow 2-subgroup of *A*_{5}. Describe the isomorphism types of a Sylow 2-subgroup and of a Sylow 3-subgroup of *A*_{6}. ## Homework 9, due Apr 5 at 12:00pm Please do the following problems from the textbook: Isaacs p67: 5.8, 5.13 and the following additional problems **A.** Prove that any group of order 7735 is cyclic. **B.** Show that *S*_{8} contains an element of order 15, but that *S*_{7} contains no such element. **C.** Find all Sylow 2- and 3-subgroups of *S*_{4}. Include a discussion of how many such subgroups there are, and the isomorphism type of each such subgroup. **D.** Find all Sylow 2-, 3-, and 5-subgroups of *S*_{5}. Include a discussion of how many such subgroups there are, and the isomorphism type of each such subgroup. ## Homework 8, due Mar 22 at 12:00pm Please do the following problems from the textbook: Isaacs p67: 5.7, 5.9, 5.10, 5.11, 5.14, 5.16 ## Homework 7, due Mar 8 at 12:00pm Please do the following problems from the textbook: Isaacs p53: 4.4 Isaacs p67: 5.1, 5.2, 5.3, 5.18 and the following additional problems **A.** Describe up to isomorphism all groups of order *pq*, where *p* > *q* are primes such that *q* does not divide *p* - 1. Hint: On 5.3, use the Sylow D Theorem. Intersecting *H* with Sylow subgroups of *G* may be helpful. Hint: On 5.18, Corollary 4.5 may be helpful. ## Homework 6, due Mar 1 at 12:00pm Please do the following problems from the textbook: Isaacs p53: 4.1, ~~4.2~~, 4.3, 4.5 and the following additional problems **A.** Explain why left multiplication doesn't give a (right) group action: i.e., explain why *x*^{g} = *gx* does not define a group action. **B.** Prove that if *G* is a finite group of order *p*^{k}, then the center *Z*(*G*) is nontrivial. Hint: consider the class equation. How big is the conjugacy class of an element of *Z*?
**C.** Recall that the *normal closure* of a group *H* is the subgroup *NC*(*H*) generated by all conjugates of *H* (i.e., *NC*(*H*) = <*H*^{g} : *g* in *G*>). Prove that if *N* is normal in *G*, and *H* is contained in *N*, then *NC*(*H*) is contained in *N*.
## Homework 5, due Feb 25 at 12:00pm Please do the following problems from the textbook: Isaacs p39: 3.4, 3.5, 3.8, 3.10, 3.12, 3.13, 3.14 and the following additional problems **A.** Describe up to isomorphism all groups of order 2*p*, where *p* > 2 is a prime. (We discussed *p* = 3 in class.) **B.** Recall that the *Frattini subgroup* of *G*, is the intersection of all maximal subgroups of *G*, and is written Phi(*G*) (with a capital Greek letter). Show that Phi( *G* / Phi(*G*)) is the trivial subgroup. Remark: On 3.4b, *H* should be taken to be a Hall pi-group. ## Homework 4, due Feb 8 at 12:00pm Please do the following problems from the textbook: Isaacs p26: 2.16, 2.18, 2.21 Isaacs p39: 3.1, 3.3, 3.9 and the following additional problems **A.** Prove Lemma 3.5 part b. (I.e., work out all of the routine details!) **B.** Show that every group of order 6 is isomorphic to either **Z**_{6} or *S*_{3}. Hint: Find the order of the subgroup generated by every element, and by every pair of elements. It may be helpful to divide into abelian and nonabelian cases. ## Homework 3, due Feb 1 at 12:00pm Please do the following problems from the textbook: Isaacs p26: 2.3, 2.8, 2.11, 2.12ab, 2.13, 2.20 and the following additional problems **A.** Show that every subgroup of the cyclic group **Z**_{n} is characteristic. **B.** If *H* is a subgroup of *G*, prove that the intersection of all conjugates *H*^{g} (over all *g* in *G*) is normal in *G*. ## Homework 2, due Jan 25 at 12:00pm *If you didn't come to office hours last week due to snow, and come this week instead, you can still get 2 bonus points.* Please do the following problems from the textbook: Isaacs p26: 2.1, 2.2, 2.4, 2.5, 2.6, 2.7 and the following additional problems **A.** Work out the details of the following: a) Show that conjugation by any *g* in *G* is an automorphism of *G* b) Show that if *H* is a subgroup of *G* and *f* is an isomorphism from *G* to *K*, then *f*(*H*) is a subgroup of *K*. c) Show that for any group *G* we have Aut(*G*) to be a group. d) Find the group structure of Aut(**Z**_{12}). **B.** Find the subgroup lattice of *Q*_{8} and *D*_{8}. Indicate the normal subgroups with diamonds, and the non-normal subgroups with circles or ovals. **C.** Construct an abelian group *A* with 8 elements such that *A* has an element of order 4, but no element of order 8. Hint on 2.5: Relate elements of G minus M to the subgroup lattice. Note on 2.6: You may assume *G* is finite -- in this case, |*G*:*H*| = |*G*| / |*H*|. This problem will be easier if you read ahead and use (2.19), (2.20), (2.21), and/or (2.22), which you should feel free to do. ## Homework 1, due Jan 18 at 12:00pm Please do the following problems from the textbook: Isaacs p12: 1.1, 1.2, 1.5, 1.6, 1.7, 1.8, 1.9 and the following additional problems **A.** We defined *D*_{2n} to consist of all rotations and flips of the *n*-gon. Prove that *D*_{2n} is a permutation group. **B.** In problem 1.7, notice that *GL*(2,2) has order 6. Is this group isomorphic to the integers mod 6? To Sym( {1,2,3} )? To neither? (Take the group operations to be multiplication on GL, addition on Z/6Z, and composition on Sym.) **C.** On this homework set (only), you can get 2 bonus points for coming by my office hours on Thursday with a question related to the course material. |