Math 3110 Spring 2010 -- Homework Problem Set #4

Due: Friday, March 19th
  1. Let \( G \) be a group such that for all \( x \in G \) we have \( x^{-1}=x \). Prove \( G \) is abelian.

  2. Which of the following are groups (If it's a group, prove it. If not, explain why not.):

    1. \( G_1 = \mathbb{Q}_{>0} \) (positive rational numbers) under multiplication.

    2. \( G_2 = (0,1] = \{ r \in \mathbb{R} \,|\, 0 < r\leq 1 \} \) (positive reals less than or equal to 1) under multiplication.

    3. \( G_3 = \mathcal{P}(X) \) (the power set of some set \( X \)) with the operation \( \cup \) (set union).

    4. \( G_4 = \{ z \in \mathbb{C} \,|\, z^4=1 \} = \{ \pm 1, \pm i \} \) where \( i = \sqrt{-1} \) under complex multiplication.

  3. Consider the Dihedral group \( D_6 \) (symmetries of a regular hexagon centered at the origin). Let \( x \) be a rotation about the origin of \( 60^{\circ} \) and \( y \) be the reflection across the \(y\)-axis. Then we know that \( D_6 = \{ 1,x,x^2,x^3,x^4,x^5,y,xy,x^2y,x^3y,x^4y,x^5y \} \) and \( x^6=1 \), \( y^2=1 \), and \( xyxy=1 \).

    1. Compute \( yx^{10}yx^{-2}y \) [Your answer should be one of the elements of \( D_6 \) listed above].

    2. Notice that \( 1A = A1 \) for all \( A \in D_6 \) (the identity commutes with everything). Are there other elements \( B \in D_6 \) such that \( AB=BA \) for all \( A \in D_6 \)? For each element either show that it commutes with everything or give an example of an element it fails to commute with.

  4. Make a Cayley tables for \( U(9) \), and \( U(5) \)
    [Remember \( U(n) = \{ k \in \mathbb{Z}_n \,|\, (k,n)=1 \} \) under multiplication mod \(n\)].

  5. For each of the following permutation: write the permutation as a product of disjoint cycles, write the permutation as a product of transpositions, and determine if the permutation is even or odd.
    1. \( (1432)(56)(254) \)

    2. \( (1234)(1423)(246) \)

    3. \( (12)(345)(1357) \)

  6. Compute \( \tau \sigma \tau^{-1} \) for each of the following pairs:
    1. \( \tau = (123) \) and \( \sigma = (12)(34) \)

    2. \( \tau = (12)(34)(56) \) and \( \sigma = (123)(456) \)

    3. \( \tau = (15)(246) \) and \( \sigma = (13245)(678) \)