10-1100/Homework Assignment 3

DBN: Classes: 2010-11: 10-1100/Navigation Additions to the MAT 1100 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 13 About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday 2 Sep 20 Tuesday - Homomorphisms and Normal Groups, Thursday - Isomorphism Theorems 3 Sep 27 Class Photo, HW1, HW1 solution 4 Oct 4 5 Oct 11 HW2, HW2 solution 6 Oct 18 7 Oct 25 Term Test 8 Nov 1 HW3, HW3 solution 9 Nov 8 Monday-Tuesday is Fall Break, One Theorem, Two Corollaries, Four Weeks 10 Nov 15 HW4, HW4 solution 11 Nov 22 12 Nov 29 HW5, HW5 solution 13 Dec 6 Boxing Day Handout, see also December 2010 Schedule. Some Class Notes F Dec 13 Final exam, Tuesday December 14 10-1, Bahen 6183 Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

This assignment is due at class time on Tuesday, November 16, 2010.

Solve the following questions

Problem 1. (Selick) Show that any group of order 56 has a normal Sylow-[math]\displaystyle{ p }[/math] subgroup, for some prime [math]\displaystyle{ p }[/math] dividing 56.

Problem 2. (Qualifying exam, May 1997) Let [math]\displaystyle{ S_5 }[/math] act on [math]\displaystyle{ ({\mathbb Z/5})^5 }[/math] by permuting the factors, and let [math]\displaystyle{ G }[/math] be the semi-direct product of [math]\displaystyle{ S_5 }[/math] and [math]\displaystyle{ ({\mathbb Z/5})^5 }[/math].

1. What is the order of [math]\displaystyle{ G }[/math]?
2. How many Sylow-5 subgroups does [math]\displaystyle{ G }[/math] have? Write down one of them.

Problem 3. (Selick) Show that the group [math]\displaystyle{ Q }[/math] of unit quaternions ([math]\displaystyle{ \{\pm 1, \pm i, \pm j, \pm k\} }[/math], subject to [math]\displaystyle{ i^2=j^2=k^2=-1\in Z(Q) }[/math] and [math]\displaystyle{ ij=k }[/math]) is not a semi-direct product of two of its proper subgroups.

Problem 4. (Qualifying exam, September 2008) Let [math]\displaystyle{ G }[/math] be a finite group and [math]\displaystyle{ p }[/math] be a prime. Show that if [math]\displaystyle{ H }[/math] is a [math]\displaystyle{ p }[/math]-subgroup of [math]\displaystyle{ G }[/math], then [math]\displaystyle{ (N_G(H):H) }[/math] is congruent to [math]\displaystyle{ (G:H) }[/math] mod [math]\displaystyle{ p }[/math]. You may wish to study the action of [math]\displaystyle{ H }[/math] on [math]\displaystyle{ G/H }[/math] by multiplication on the left.

Problem 5. (easy)

1. Prove that in any ring, [math]\displaystyle{ (-1)^2=1 }[/math].
2. Prove that even in a ring without a unit, [math]\displaystyle{ (-a)^2=a^2 }[/math].

(Feel free to do the second part first and then to substitute [math]\displaystyle{ a=1 }[/math]).

Problem 6.

1. (Qualifying exam, April 2009) Prove that a finite integral domain is a field.
2. (Qualifying exam, September 2008) Prove that in a finite commutative ring, every prime ideal is maximal.

Problem 7. (Dummit and Foote) A ring [math]\displaystyle{ R }[/math] is called a Boolean ring if [math]\displaystyle{ a^2=a }[/math] for all [math]\displaystyle{ a\in R }[/math].

1. Prove that every Boolean ring is commutative.
2. Prove that the only Boolean ring that is also an integral domain is [math]\displaystyle{ {\mathbb Z}/2 }[/math].