A matrix $a\in GL_{n}(F)$ is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices.
I have already proved here that the following are equivalent
- $A\in N$
- there exist a non singular $D$ (diagonal matrix ) and a permutation matrix $P$ such that $A=DP$
- there exist a non singular $D$ (diagonal matrix ) and a permutation matrix $P$ such that $A=PD$
My aim is to show the following
1) $N$ is a subgroup of $G=GL_{n}(F)$.
2) $T=B\cap N$ is the subgroup of $G$ consisting of all diagonal matrices.
3) $N$ is a semi direct product of $T$ and $W$, here $W$ is group of all permutation matrices.
4) $N=N_{G}(T)$, normalizer of $T$ in $G$.
5) $N_{G}(T)/T$ is generated by finite set $S$ of involutions (elements of degree 2) and $|S|=n-1$.
Using the above mentioned lemma, I have proved 1, 2 and 4 is already solved here.
I have made an attempt to solve question 3. Please correct me if my reasoning for question 3 is wrong.
Now by the lemma, we conclude $N=WT$ also $W\cap T$ is trivial, so now only thing left to show is that $T$ is normal in $N$, but it is obvious from question 4 (since $N_{G}(T)=N$, restricting it to $N$, $N_{G}(T)|_{N}=N$ hence $T$ is normal subgroup). So $N$ is indeed a semidirect product of $T$ and $W$.
But I am not able to solve question $5$, Kindly help.