Math 3110 Fall 2011 -- Homework Problem Set #4

Due: Monday, October 24th

Chapter 6

  1. Cayley's Theorem tells us that \( D_4 \) is isomorphic to a subgroup of \( S_8 \). Find the subgroup of \( S_8 \) which corresponds to the ordering: \( R_0, R_{90}, R_{180}, R_{270}, H, V, D, D' \) (i.e. \( R_0 \) is element 1, \( R_{270} \) is element 4, \( D' \) is element 8, etc.)

  2. The following pairs of groups are either isomorphic or not. If the groups are isomorphic, prove it. If the groups are not ismorphic, show why they are not.

    1. \( (\mathbb{Z}_2)^{2 \times 2} \) and \( \mathrm{GL}_2(\mathbb{R}) \)

    2. \( \mathbb{Z}_{100} \) and \( D_{50} \)

    3. \( U(7) \) and \( \mathbb{Z}_6 \)

    4. \( A_4 \) and \( D_6 \)

    5. \( \mathbb{R}^* \) and \( \mathbb{C}^* \) (non-zero reals and complexes under multiplication)

    6. \( H = \left\{ \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \,{\Huge|}\, n \in \mathbb{Q} \right\} \) and \( \mathbb{Q} \)
      [Note: \( H \) is a subgroup of \( \mathrm{GL}_2(\mathbb{Q}) \).]


  3. Let \( \varphi:G_1 \rightarrow G_2 \) be an isomorphism.

    1. Show \( G_1 \) is abelian if and only if \( G_2 \) is abelian.

    2. Let \( K \) be a subgroup of \( G_2 \). Show \( \varphi^{-1}(K) = \{ g \in G_1 \,|\, \varphi(g) \in K \} \) is a subgroup of \( G_1 \).