Let $A$ and $B$ be any invertible $4 \times 4$ matrices with $0$ and $1$ everywhere, and let $H=\{A^n| n \in \mathbf{Z}\}$, $N=\{A^nB^m | n,m\in \mathbf{Z} \}$
- are the subsets $H$ and $N$ subgroups of the general linear group $\text{GL}(n,\mathbf{R})$?
- what is the order of $AB$?
I wrote two matrices for $A$ and $B$, and letting $n≤3$,and $m≤3$ so that $H$ consists of $A$, square of $A$ and the cube of $A$. I found that $PQ$ is not in $H$ for some $P$ and $Q$ in $H$ , there I want to conclude that $H$ may not be a subgroup of $\text{GL}(n,\mathbf{R})$. I request some help here. Can the order of $AB$ be written in terms of $n$ and $m$?