Math 4720 & 5210 Fall 2010 -- Homework Problem Set #1

Due: Tuesday, September 7th
  1. Let \( S \) be a non-empty set and \( F(S) \) the free group generated by \( S \). Prove that \( a \) has infinite order for all \( a \in S \).
    [This implies that all free groups (except the trivial one) are infinite.]

    5210 Students: Suppose \( S = \{ a,b,c \} \). Show \( ab^{-3}a^2c \) has infinite order.

    (Optional) Challenge: Prove that the identity is the only element of finite order in a free group.

  2. Find a set of generators and relations for \( S_4 \).

  3. Let \( A \in \mathrm{GL}_n(R) \). If \( A^{-1} \) exists, find it. If not, explain why it does not exist.

    1. \( \displaystyle{ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} } \) and \( R = \mathbb{Z}_9 \).

    2. \( \displaystyle{ A = \begin{bmatrix} x^2+1 & x^2-2x+1 \\ x & x-2 \end{bmatrix} } \) and \( R = \mathbb{R}[x] \) (polynomials with real coefficients).

    3. \( \displaystyle{ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 2 \\ 3 & 2 & 1 \end{bmatrix} } \) and \( R = \mathbb{Z}_{12} \).

    4. 5210 Students: \( \displaystyle{ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 4 & 2 \end{bmatrix} } \) and \( R = \mathbb{Z}_{11} \).

  4. Let \( A \) and \( B \) be groups.

    1. Prove \( A \times B \cong B \times A \)

    2. Prove \( A \times B \) is abelian if and only if both \( A \) and \( B \) are abelian.

  5. Actions on Regular Polyhedra (See https://en.wikipedia.org/wiki/Regular_polyhedron and problems 20-23 in 1.7)

    1. Prove the group of rigid motions (rotations) of a tetrahedron is isomorphic to a subgroup of \( S_4 \).

    2. Identify this subgroup (either give me its name or write down its elements).

    3. 5210 Students: Prove the group of rigid motions (rotations) of a cube is \( S_4 \).