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I have a subgroup of $GL(3, \mathbb{R})$, given by

\begin{pmatrix} a & 0 & b \\ 0 & a & c\\ 0 & 0 & d \end{pmatrix}

with a,d different from 0 and a,b,c and d real numbers. Let's call this subgroup G.

Now, we consider the following subgroups of G:

\begin{pmatrix} 1 & 0 & b \\ 0 & 1 & c\\ 0 & 0 & 1 \end{pmatrix}

\begin{pmatrix} a & 0 & 0 \\ 0 & a & 0\\ 0 & 0 & d \end{pmatrix}

Let's call N the first one and Q the second one. Now, I've proven that N is a normal subgroup of G, but now I have to show that G/N is isomorphic to Q and decide if G is isomorphic to $N \times Q$. I suppose that I have to use the isomorphism theorems for groups. Since I have N normal subgroup and Q, then I've been trying to use the second one with no success.

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    **Hint** For any $\pmatrix{a&0&b\\0&a&c\\0&0&d} \in G$ and $n \in N$, we have $gn = \pmatrix{a&*&*\\0&a&*\\0&0&d}$ for some entries $*$. – 2017-01-19
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    N, Q are abelian – 2017-01-22

1 Answers 1

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For technical reasons I will represent the elements of $N$ as $\begin{pmatrix} 1 & 0 & x\\0 & 1 & y\\0 & 0 & 1\end{pmatrix}$ and the elements of $Q$ as $\begin{pmatrix}u & 0 & 0\\0 & u & 0\\0 & 0 & v\end{pmatrix}$. If we take an arbitrary element $g \in G$ defined by some $a, b, c$ and $d$ and consider its left coset $gn$, where $n$ runs through $N$ we see that these elements are of the form \begin{pmatrix}a & 0 & a x + b\\0 & a & a y + c\\0 & 0 & d\end{pmatrix}. For the particular choice of $x = -b/a$ and $y = -c/a$ we see that this element belongs to $Q$ with $u = a$ and $v = a$. We have shown that $\overline{g} \cap Q = \{q\}$. In other words, we can take for every element of the quotient (a left coset) a representative in $Q$. This shows that $G = N \rtimes Q$, the semidirect product of $N$ and $Q$.