Home Syllabus Schedule Links **Homework** | | | | # Homework Starred* problems should be completed if you are enrolled in the course at the 6000 level. I recommend (but do not require) that all students read and attempt these problems. If you make some significant progress, you should turn it in. ## Homework 12, due never Isaacs p81: 7.1, 7.2, 7.10 **A.** Explain why the quaternion group *Q*_{8} does not split nontrivially as a semidirect product. **B.** Show how the symmetric group *S*_{4} splits as a semidirect product over its normal subgroup *N* of order 4. Be sure to describe the action of the complement *H* on *N*. ## Homework 11, due Apr 27 at 12:00pm Isaacs p81: 6.3, 6.4, 6.5, 6.7, 6.10, 6.12* **A.** Show that a simple group of order 168 has subgroups of order 21, but no elements of order 21. ## Homework 10, due Apr 15 at 12:00pm Isaacs p67: 5.10, 5.13, 5.14, 5.18, 5.20* Isaacs p81: 6.8 ## Homework 9, due Apr 1 at 12:00pm Isaacs p67: 5.1, 5.2, 5.3, 5.7, 5.9, 5.12 ## Homework 8, due Mar 25 at 12:00pm Isaacs p53: 4.3, 4.5 **A.** Suppose |*G*| = *p*^{a} for a prime *p* and positive integer *a*. Show that *Z*(*G*) is nontrivial. Hint: Considering fixed points under the conjugation action, and examine the class equation mod *p*. **B.** The action of Sym [*n*] on [*n*] = {1, 2, ..., *n*} induces an action on subsets of [*n*], where {*x*_{1}, *x*_{2}, ...}˙σ = {*x*_{1}˙σ, *x*_{2}˙σ, ...}. Find the index and order of the stabilizer of a subset *S* of [*n*]. **C.** Let *G* be a finite group. Show that [*H* : *H*∩*K*] ≤ [*G* : *K* ], and that equality holds if and only if *G* = *HK*. **D.** Let *p* be a prime, and *G* be a finite group. Show that if [*G* : *H*] = *p*, and *P* is a Sylow *p*-subgroup of *G*, then *HP* = *G*. Hint: Problem C may be helpful for Problem D. ## Homework 7, due Mar 18 at 12:00pm Isaacs ~~p39~~p53: 4.1, 4.4 **A.** Let *G* be a finite group and *N* be a normal subgroup contained in Φ( *G* ). Show that Φ( *G*/*N* ) = Φ( *G* ) / *N*. **B.** Find a group *G* such that Φ( *G* ) is isomorphic to the cyclic group **Z**_{4}, but show that there is no group *K* such that *K*/Φ( *K* ) is isomorphic to **Z**_{4}. **C.** Suppose that *G* is a group of order 60, and *H* is a subgroup of *G* having order 15. Show that *G* is not a simple group.
Note: There does exist a simple group *S* having order 60. By your argument for this problem, *S* has no subgroups of order 15. **D.** Let *V* be an *n*-dimensional vector space over a field *F* with (a finite number) *q* elements. We previously calculated the order of *G* = *GL*(*V*). For any nonzero *v* in *V*, find the order of Stab_{G} *v*. ## Homework 6, due Mar 4 at 12:00pm Isaacs p39: 3.6, 3.8, 3.9, 3.10, 3.12 **A.** i. Show that if *G* is a permutation group on *X*, then *G* has a naturally defined group action on *X*. ii. Explain why left conjugation ( *H*⋅*g* = *gHg*^{-1} ) does not define a group action on *L*(*G*) as defined in Isaacs, but right conjugation does. ## Homework 5, due Feb 18 at 12:00pm Isaacs p26: 2.9, 2.16 Isaacs p39: 3.1, 3.3, 3.4, 3.5* **A.** If *H* is any subgroup of *G*, prove that the subgroup generated by all conjugates of *H* is normal in *G*. That is, prove that <*H*^{g} : *g* ∈ *G*> ◅ G ## Homework 4, due Feb 11 at 12:00pm Isaacs p26: 2.1, 2.6, 2.8, 2.10*, 2.17, 2.18, 2.21 ## Homework 3, due Feb 4 at 12:00pm Isaacs p26: 2.3, 2.5, 2.7, 2.12, 2.13*, 2.19, 2.20 ## Homework 2, due Jan 28 at 12:00pm Isaacs p12: 1.8 Isaacs p26: 2.2, 2.4 **A.** Prove that if *H* is a nonempty finite subset of a group *G* that is closed under multiplication, then *H* is a subgroup of *G*. **B.** For each of the following subsets of *GL*_{n}(**F**) (where **F** is any field), either prove or disprove that it forms a subgroup: i) The set of all matrices that are zero except on the diagonal. ii) The set of all matrices with determinant 1. iii) The set of all (weakly) upper triangular matrices. **C.** Let *m* and *n* > 1 be relatively prime integers. Show that if *G* is an abelian group with elements of order *m* and of order *n*, then *G* has an element of order *mn*. Conclude that any abelian group of order 6 is cyclic. ## Homework 1, due Jan 21 at 12:00pm Isaacs p12: 1.1*, 1.2, 1.5, 1.6, 1.7, 1.9 **A.** For the triangular bipyramid obtained by gluing two regular tetrahedra together along a face, describe the rotation group, and the congruence group. |