Math 3110 Spring 2010 -- Homework Problem Set #5

Due: Monday, March 29th
  1. Workin' Mod 75.

    1. Draw the subgroup lattice of \( \mathbb{Z}_{75} \).

    2. Make a table which lists the orders of the elements in \( \mathbb{Z}_{75} \) along with how many elements have that order.

  2. Show that \( \displaystyle{ H = \left\{ \begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix} \,\Big|\, r \in \mathbb{R} \right\} } \) is a subgroup of \( \mathrm{GL}_2(\mathbb{R}) \). Don't forget to make sure \( H \) is a subset of \( \mathrm{GL}_2(\mathbb{R}) \) first!

  3. Let \( G \) be a group and \( a \in G \). Let \( C_a = \{ x \in G \,|\, ax=xa \} \) --- this is the set of everything in \( G \) that commutes with \( a \). \( C_a \) is called the centralizer of \( a \) in \( G \). Let \( Z(G) = \{ x \in G \,|\, gx=xg \mbox{ for all } g \in G \} \) --- this is the set of elements of \( G \) which commute with everything in \( G \). \( Z(G) \) is called the center of \( G \). For example: In the last homework we showed that \( Z(D_6) = \{ 1,x^3 \} \).

    1. Show that \( C_a \) is a subgroup of \( G \).

    2. Show that \( Z(G) \) is a subgroup of \( G \).

    3. Show that \( \displaystyle{ Z(G) = \bigcap_{g \in G} C_g } \).

  4. List all of the cyclic subgroups of \( D_8 = \{ 1,x,\dots,x^7,y,xy,\dots,x^7y \} \).

  5. Let \( G \) be a group with identity \( e \) and let \( a,b \in G \) be commuting elements (i.e. \( ab=ba \)) whose orders are finite. Also, suppose that \( \langle a \rangle \cap \langle b \rangle = \{ e \} \). Show that the order of \( ab \) is the least common multiple of the orders of \( a \) and \( b \). [Problem #20 on page 77 has the definition of least common multiple.]