MSU MATHEMATICS |
MTH 418H
Honors Algebra I |
| Fall 2005 |
- Instructor: Prof. Peter Magyar, magyar@math.msu.edu, tel. 353-6330, Wells Hall D-326.Map
Office hours: Mon, Wed, Fri 10am--12:00pm, and by appointment.- Course Information: Grading, exam, and homework policies. Syllabus.
- Announcements
- Final Exam: Thurs Dec 15, 12:45--2:45pm, in the usual room Wells C-215. The exam will cover all the material of the course, with about 1/3 devoted to each of:
- Test 1 -- integers, rational numbers, reals, polynomial rings (see Review HW)
- Test 2 -- complex numbers and functions (see Review HW)
- Group theory (see HW 12/2 and 12/9)
- Review session: Tentatively Tues Dec 13, 11-12:30 Wells A117. I will discuss the Review HW and HW 12/2 & 12/9. Those who have not done an oral quiz presentation should hand in one problem from these assignments. This and all extra credit work should be turned in before the end of the day (in class or put under my door, Wells D-326).
- Homework: I will not collect homework, but it will be the basis of the weekly quizzes (usually on Fridays). You may hand in Extra Credit problems at any time during the course, no matter when they were posted. Exercise and page numbers refer to the Artin text.
- Old Homework
- HW 11/23: No Quiz. Text: Ch 2.1. Notes 11/18, 11/23.
- Write out all 6 elements of the group D3 , the symmetries of the triangle, as pictures, as permutations of the 3 corners, and also in terms of letters: {ι, β, β², α, αβ, αβ² },where ι is the identity symmetry, α is a reflection and β is a rotation.Compute the 6×6 multiplication table. (It is neatest to write the table in terms of the letter notation.)
- Do the same for the symmetries of the square, the group D4 ⊂ S4 . Prove that D4 contains only 8 symmetries of the square, but S4 contains 4! = 24 permutations of {1,2,3,4}.
- Extra Credit: Do the above for the symmetries of a regular n-gon, which can be considered as permutations of the n vertices: D4 ⊂ S4 . Identify the permutations belonging to D4 , and find a convenient notation in terms of letters. Write the multiplication table.
- Extra Credit: Draw an object in the plane whose symmetry group contains only rotation symmetries, with no reflection symmetries. Next, find an object whose only symmetries are the n rotations by angles 2πk/n for k = 0,1,2,...,n-1.
- HW 12/2: Quiz on groups. Text: Ch 2.2, 2.3, 2.8. Notes 11/31.
- Consider (Z , +), the abelian group of integers under the addition operation. Show that the only subgroups of Z are nZ, the multiples of some integer n. That is, suppose H is an arbitrary subgroup, meaning that if α,β ∈ H , then α˙β , α-1 ∈ H . Then give a careful proof that H = nZ where n = min{ k > 0 , k ∈ H }.
- Subgroups of Dn
- Show that the dihedral group D3 contains exactly five proper subgroups: the trivial group {ι}, the 3-element group of rotations, and three 2-element groups generated by reflections. Hint: Any subgroup containing both a rotation and a reflection must be all of D3 .
- Determine all the subgroups of the dihedral group D6 .
- Determine all the subgroups of an arbitrary Dn . Hint: each subgroup is the symmetry group of a decorated n-gon.
- The order of a group means the number of elements in it, whereas the order of an element g ∈ G is the smallest k > 0 with gk = 1 .
- Prove that gk = 1 if and only if ord(g) | k (that is, the order of g divides k).
- Show that for any group G of order n, we have gn = 1 for any g ∈ G . What does this mean about the order of the element g ? (Not just ord(g) < n.)
- Let (Zp , + , ×) be the field of clock arithmetic modulo a prime p, and let (Zp× , ×) denote the group of non-zero elements of Zp under the multiplication operation.
- Check that Zp× is indeed a group. (Indeed, for any field F, there is a corresponding multiplicative group F×.)
- Example: Write the multiplication table of Z5×, and show that it is isomorphic to the cyclic group C4 .(Extra Credit: Prove Zp× is isomorphic to Cp-1 for every prime p.)
- Show that xp-1 = 1 for any x ∈ Zp×.
- Fermat's Little Theorem: For any integer n and any prime p, we have p | np - n . Example: 25 - 2 = 30 , a multiple of 5. Prove the theorem using the above group properties of Zp×.
- Wilson's Theorem: For any prime p, we have p | (p-1)! + 1 . Example: (5-1)! + 1 = 25 , a multiple of 5. Prove the theorem using the above group properties of Zp×.
- HW 12/9: Quiz on cosets and normal subgroups. Text: Ch 2.6, 2.10. Notes 12/9.
- Recall that for any subgroup H of G , the quotient G/H is the collection of all cosets gH, where g ∈ G. In general, if we try to define a product on G/H by:g1H•g2H := { g1h1g2h2 where h1, h2 ∈ H } ,the result is not always another coset, so the multiplication is not closed.
Show that if H is normal, meaning gHg-1 = H, or equivalently gH = Hg, then we have g1H • g2H = g1g2H ∈ G/H , so that the multiplication is closed. Furthermore, show thisoperation defines a group (associative, with identity, with inverses). - Let Cn = {ι, x, x², ... , xn-1} with xn = 1 .
- Show that if gcd(j,n) = 1 , then the cyclic subgroup < xj > is all of Cn .
- Show that for any x ∈ G, we have: ord(xj) = n/gcd(j,n).
- Show that if gcd(m,n) = 1, then Cm × Cn is isomorphic to Cmn .
- Let G be a group with mn elements, generated by the elements a, b with relations am = bn = ι and ab = ba . Show that G is isomorphic to the product group Cm × Cn .
- The quaternion group consists of 8 elements denoted: Q = {±1, ±i, ±j, ±k} with the relations described on p. 48. Prove that Q is not isomorphic to the other non-abelian 8-element group D4 .
- Determine which subgroups of Dn are normal.
- Some more problems from the text, pp 69--77.
- Ex 2.2.17: abelian group.
- Ex 2.3.1: a surprising isomorphism
- Ex 2.4.13: definition of normal subgroup
- Consider the following groups of order 8: the abelian groups C8 , C4 × C2 , C2 × C2 × C2 ; the dihedral group D4 ; and the quaternion group Q .
- Show that none of these groups are isomorphic to each other.
- Show that any group of order 8 is isomorphic to one of these.
- Handouts:
2004, Groups