Problem 5.4.14 (Group of upper triangular matrices all of whose diagonal entries are equal)

Question

Let $G = \{(a_{ij}) \in \mathrm{GL}_n(F) \mid a_{ij} = 0 	ext{ if } i>j, 	ext{ and } a_{11} = a_{22}= \cdots = a_{nn}\}$, where $F$ is a field, be the group of upper triangular matrices all of whose diagonal entries are equal. Prove that $G \simeq D 	imes U$, where $D$ is the group of all nonzero multiples of the identity matrix and $U$ is the group of upper triangular matrices with $1$'s down the diagonal.

Richard Ganaye · Accepted Answer

\begin{proof} 
\begin{itemize}
\item First $G = DU$: if $M = (a_{ij}) \in G$, then $a_{11} = a_{22}= \cdots = a_{nn} = a$ for some $a \in F$, and
$$M = (a I_n) (a^{-1} M),$$
where $a I_n \in D$ and $a^{-1} M \in U$.

\item $D \cap U = \{I_n\}$: if $M  \in D \cap U$, then $M = aI_n$ for some $a \in F$, and $a = 1$ since $M \in U$, so $M = I_n$.

\item Every element $M = aI_n$ of $D$ commutes with every element $N$ of $U$.

If $P =MN$ is any element of $G = DU$, where $M \in D$ and $N \in U$, then for every $V \in U$,
$$ P V P^{-1} = N^{-1} (M^{-1} V M) N = N^{-1} V N \in U,$$
so $U \unlhd G$, and similarly $D \unlhd G$.
\end{itemize}
By Theorem 9,
$$G = DU \simeq D 	imes U.$$
\end{proof}

Problem 5.4.14 (Group of upper triangular matrices all of whose diagonal entries are equal)

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Problem 5.4.14 (Group of upper triangular matrices all of whose diagonal entries are equal)

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