Week 8 Problems

These exercises are taken from the text, Section 10.4.

  1. (Problem 4) Let $T$ be the group of nonsingular upper triangular $2\times 2$ matrices with entries in $\mathbb{R}$. In other words, \(T = \{\left(\begin{matrix} a & b \\ 0 & c\end{matrix}\right) : a,b,c\in\mathbb{R}, ac\not=0\}.\) Let $U$ be the subset of matrices where $a=c=1$.
    • Prove that $U$ is a subgroup of $T$.
    • Prove that $U$ is abelian.
    • Prove that $U$ is normal in $T$.
    • Show that $T/U$ is abelian.
    • Is $T$ normal in $\mathrm{GL}_{2}(\mathbb{R})$?

  2. (Problems 6/7 ) Prove or disprove:

    • If $G$ is a group with normal subgroup $H$, and both $G/H$ and $H$ are abelian, then $G$ is abelian.
    • If $G$ is an abelian group with normal subgroup $H$, then both $G/H$ and $H$ are abelian.

  3. (Problem 11) Let $G$ be a group with exactly one subgroup $H$ containing $k$ elements. Prove that $H$ is a normal subgroup of $G$.

  4. (Problem 12) Let $G$ be a group and $g\in G$ an element of $G$. Define the centralizer of $g$ in $G$ to be the set \(C(g) =\{ x\in G : xg = gx\}.\)

    • Show that $C(g)$ is a subgroup of $G$.
    • Show that if the subgroup $\langle g \rangle$ of $G$ generated by $G$ is normal, then so is $C(g)$.

  5. (Problem 14) Let $G$ be a group and let $G’$ be the subgroup of $G$ consisting of all finite products of elements of the form $aba^{-1}b^{-1}$ where $a,b\in G$.

    • Show that $G’$ is a normal subgroup of $G$.
    • Let $N$ be a normal subgroup of $G$. Prove that $G/N$ is abelian if and only if $N$ contains $G’$.

$G’$ is called the commutator subgroup of $G$.