Math 3110 Fall 2011 -- Homework Problem Set #1

Due: Wednesday, September 7th
  1. For each of the following sets \( G \) equipped its given operation, show \( G \) is not a group. Then explain which group axioms do hold.

    1. \( G = \{ 3m+2 \,|\, m \in \mathbb{Z} \} = \{ \dots, -4, -1, 2, 5, \dots \} \) with addition.

    2. \( \mathbb{R}_{<0} \) (negative real numbers) with multiplication.

    3. \( \mathbb{Q} \) (rational numbers) with subtraction.

    4. \( G = \{ x \in D_4 \,|\, x \mbox{ is a reflection} \} \) with function composition.


  2. Write down the Cayley table for \( D_5 \) (symmetries of a regular pentagon).

    Use the given labels: \(p,q,r,s,t \) for reflections and \( 1,x,x^2,x^3,x^4 \) for the (counter clockwise) rotations. In your table, order the elements as follows: \( 1 \; x \; x^2 \; x^3 \; x^4 \; p \; q \; r \; s \; t \)

    Alternate Approach: For those brave enough to try the generator and relation approach, you may list elements as \(1 \; x \; x^2 \; x^3 \; x^4 \; y \; xy \; x^2y \; x^3y \; x^4y \) and use the relations: \( x^5=1 \), \( y^2=1 \), and \( (xy)^2=1 \).

  3. (Chapter #2 number 16): Let \( a,b \in G \) where \( G \) is a group. Prove that \( (ab)^{-1} = b^{-1}a^{-1} \). Then find an example (a group and two elements \(a \) and \(b\) ) to show that it is possible for \( (ab)^{-2} \not= b^{-2}a^{-2} \). Finally, find distinct non-identity elements in some non-Abelian group such that \( (ab)^{-1} = a^{-1}b^{-1} \).

  4. (Chapter #2 number 26): Let \( G \) be a group such that \( (ab)^2=a^2b^2 \) for all \( a,b \in G\). Show that \( G \) is Abelian.