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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.