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Course: MATH 3106, Fall 2009
School: ECCD
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FOR PROBLEMS TUTORIAL 5 AND ASSIGNMENT 5. Problem 1. Find the commutator subgroup Q of the quaternion group Q. Problem 2. (2) Find all (3) Find all (4) Find all (1) Compute the order |g| for all g D4 . conjugacy classes in D4 . normal subgroups of D4 . homomorphic images of D4 . Problem 3. Draw the lattice diagram for G if G (1) = S3 . (2) G = D4 . Defend your answer! Problem 4. Show that D4 is not isomorphic...

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FOR PROBLEMS TUTORIAL 5 AND ASSIGNMENT 5. Problem 1. Find the commutator subgroup Q of the quaternion group Q. Problem 2. (2) Find all (3) Find all (4) Find all (1) Compute the order |g| for all g D4 . conjugacy classes in D4 . normal subgroups of D4 . homomorphic images of D4 . Problem 3. Draw the lattice diagram for G if G (1) = S3 . (2) G = D4 . Defend your answer! Problem 4. Show that D4 is not isomorphic to Q. Problem 5. Show that C /C0 R+ , where R+ denotes the = group of positive real numbers under multiplic...

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ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Tutorial 1, Sep 18 Example 1. Show that if p &gt; 2 is a prime number and a = 0(mod p), then the equation x2 a2(mod p) has exactly two solutions in Z. p Example 2. Un = {z1, z2, . . . , zn} C is a cyc
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Tutorial 2, Sep 25 Example 1. Prove that Sn =&lt; (12), (13), . . . , (1n) &gt;, n 3. Example 2. Prove that the two cycles (1352) and (2461) are conjugate in S8. Example 3. Prove that the two permutations
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Tutorial 3, Oct 2 Claim 1. Let G be a group and g G be an element of G. Recall that CG(g) = {a G | ag = ga} is the centralizer of g in G. Then xgx-1 = ygy -1 x-1y CG(g). Claim 2. Let X be a gener
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Tutorial 4, Oct 16 Claim 1. A4 has no subgroup of order 6 (hence the converse of Lagrange's theorem is false). Recall that the inner automorphism a corresponding to a G is defined by the equality a(
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 1Due date: Monday, September 18 at 13:30. Problem 1. Let N denote the set of all positive integer numbers, and let : N N N be the binary operation defined by : (m, n) mn . Prove that
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 2Due date: Monday, September 25 at 13:30.Problem 1. Invertible elements and zero divisors in Zn . (1) Find the inverses of and in Z7 . 2, 4 6 (2) Find all the invertible elements and
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 3Due date: Monday, October 2 at 13:30.Problem 1. Generators for Sn and for An . Let n 3. (1) Prove that Sn =&lt; (12), (123 . . . n) &gt;. (2) Prove that An =&lt; (123), (124), . . . , (12n) &gt;.
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 4Due date: Monday, October 16 at 13:30.Problem 1. Find a subgroup of S4 isomorphic to the group (a) Z3 ; (b) K4 . Problem 2. Show that GL2 (R) is not isomorphic to the multiplicative gr
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 6Due date: Thursday, November 9 at 11:30.Problem 1. Let G be a group, and let H G and K G be such that H K = {1}. Define : H K HK by (h, k) = hk. (1) Show that is a homomorphism of
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 7Due date: Monday, November 13 at 13:30.Problem 1. Find all homomorphic images of D3 , up to isomorphism. Problem 2. Let K G, and let H be a subgroup of G so that K H. Show that if H G
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 8Due date: Monday, November 20 at 13:30.Problem 1. Let G be a group acting on a set X. (1) Define a relation on X as follows: let x, y X, we write x y (mod G) if there is g G so that
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 9Due date: Monday, November 27 at 13:30.Problem 1. If p and q are primes, q &lt; p and n &gt; 0 is an integer, show that no group of order qpn is simple. Problem 2. 8.3 23(b,c) (page 443). Pr
ECCD - MATH - 3106
INTRODUCTION TO GROUP THEORY MATH 3106, FALL 2006Assignment 10Due date: Monday, December 4 at 13:30. The following problems are to be submitted: 1(1),1(2),2(1),2(2), 3 and 4(1).Problem 1. Let G be a finite group, p be a prime divisor of |G|, and
ECCD - MATH - 3106
Introduction to Group Theory Math 3106, Fall 2006December 6, 2006 Assignments 7 and 8. Partial solution.Assignment 7Problem 3. (1) We know that S4 /V4 S3 (see solutions to the test and first assignments), and so = S4 /V4 = {V4 , (12)V4 , (13)V4
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Introduction to Group Theory Math 3106 Partial Solutions to Assignments 1 - 5 Problem 1. Prove Wilson's Theorem: If p is a prime, then (p-1)! -1 mod p. Solution. As p is a prime, Z = Zp \ {0}, and so P = (p - 1)! is the p product of all elements of
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