Abstract Algebra: Theory and Applications

For each of the following groups $G\text{,}$ determine whether $H$ is a normal subgroup of $G\text{.}$ If $H$ is a normal subgroup, write out a Cayley table for the factor group $G/H\text{.}$

Let $T$ be the group of nonsingular upper triangular $2\times 2$ matrices with entries in $\mathbb{R}\text{;}$ that is, matrices of the form

where $a\text{,}$ $b\text{,}$ $c\in \mathbb{R}$ and $ac\ne 0\text{.}$ Let $U$ consist of matrices of the form

where $x\in \mathbb{R}\text{.}$

Define the centralizer of an element $g$ in a group $G$ to be the set

Show that $C(g)$ is a subgroup of $G\text{.}$ If $g$ generates a normal subgroup of $G\text{,}$ prove that $C(g)$ is normal in $G\text{.}$

Recall that the center of a group $G$ is the set

Let $G$ be a group and let $G^{\prime }=⟨ab{a}^{-1}{b}^{-1}⟩\text{;}$ that is, $G^{\prime }$ is the subgroup of all finite products of elements in $G$ of the form $ab{a}^{-1}{b}^{-1}\text{.}$ The subgroup $G^{\prime }$ is called the commutator subgroup of $G\text{.}$