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MATH 4081 - Abstract Algebra - Group Activity 2 - Fall Semester 2015 Dr. Brandon Samples - Department of Mathematics - Georgia College Group Activity 2: Due 12/08/2015 Directions: The following questions are to be completed by your group members. I want to put an emphasis on the math above the typing here, so you do not have to type your solutions. I also want the labor of writing to be shared. In other words, work together and divide up the writing. Do not however divide up the thinking. 1. Recall that a subgroup H ⊂ G is called normal if gHg −1 ⊂ H for all g ∈ G. Prove that the set {e, ρ2 } is a normal subgroup of the square group Sq. S. To prove that H is a normal subgroup, we must compute gHg −1 for all elements g ∈ G. Consider the following: eHe = {eee, eρ2 e} = H ρ1 Hρ3 = {ρ1 eρ3 , ρ1 ρ2 ρ3 } = H ρ2 Hρ2 = {ρ2 eρ2 , ρ2 ρ2 ρ2 } = H ρ3 Hρ1 = {ρ3 eρ1 , ρ3 ρ2 ρ1 } = H FA HFA = {FA eFA , FA ρ2 FA } = H FB HFB = {FB eFB , FB ρ2 FB } = H FC HFC = {FC eFC , FC ρ2 FC } = H FD HFD = {FD eFD , FD ρ2 FD } = H 2. We can define a group G/H called a quotient group whenever H ⊂ G is normal. The group operation for the elements gH ∈ G/H is given by (g1 H) ∗ (g2 H) = g1 ∗ g2 H. a. Build the multiplication table for G/H with H and G given in part 1. S. ◦ eH ρ1 H FA H FC H eH eH ρ1 H FA H FC H ρ1 H ρ1 H eH FC H FA H FA H FA H FC H eH ρ1 H FC H FC H FA H ρ1 H eH b. Verify that G/H is a group. S. It’s clear that eH is the identify element and associativity follows from the fact that composition is an associative operation. The diagonal with slope negative one shows that every element is its own inverse. In fact, it’s abelian! c. To what group from Test 2 is G/H isomorphic? S. This group is isomorphic to the so-called Klein four group. It’s the group on exam two that is not Z4 with operation of addition. 3. Suppose φ : G → K is a group homomorphism. Prove that Im(φ) = {y = φ(x) ∈ K | x ∈ G} is a subgroup of K. S. Proof: Let y ∈ Im(φ). Since eK = φ(eG ) = φ(y ∗ y −1 ) = φ(y) ? φ(y −1 ), it follows that φ(y −1 ) = (φ(y))−1 . The former is in the image since y −1 ∈ G and the equality shows that Im(φ) is closed under inverse. Next, consider two elements y1 , y2 ∈ Im(φ). Since y1 ? y2 = φ(x1 ) ? φ(x2 ) = φ(x1 ∗ x2 ) and x1 ∗ x2 ∈ G, we see that y1 ? y2 ∈ Im(φ). This establishes closure under the inherited ? operation from K. Q.E.D. 4. Use Burnside’s Theorem to determine the number of distinct ways to place four blue, two green, and two yellow chairs around a circular table. S. In the interest of time, I at least offer the answer is 33. 5. Use Burnside’s Theorem to determine the number of distinct ways to paint the faces of a cube with three blue faces, one green face, and two yellow faces. S. In the interest of time, I at least offer the answer is 3.