Carl Dahlquist

Let F be a group where $F=\{A \in GL_3( \Bbb{Z}_2)\}$.

F is associative.

F contains the identity.

F contains inverses.

One subgroup of this matrix is:

(2)
\begin{align} H=\{\left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right], \left[\begin{array}{ccc}1 & 1 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\} \end{align}

The subgroup is closed.

Inverses are present.

Identity exists.

Left cosets:
A few cosets of this subgroup are:

(6)
\begin{align} \left[\begin{array}{ccc}1 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\cdot H =\{\left[\begin{array}{ccc}1 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right], \left[\begin{array}{ccc}1 & 1 & 1 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\} \end{align}
(7)
\begin{align} \left[\begin{array}{ccc}1 & 0 & 0 \\1 & 1 & 0 \\0 & 0 & 1\end{array}\right]\cdot H =\{\left[\begin{array}{ccc}1 & 0 & 0 \\1 & 1 & 0 \\0 & 0 & 1\end{array}\right], \left[\begin{array}{ccc}1 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{array}\right]\} \end{align}
(8)
\begin{align} \left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 1\end{array}\right]\cdot H =\{\left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 1\end{array}\right], \left[\begin{array}{ccc}1 & 1 & 0 \\0 & 1 & 0 \\1 & 1 & 1\end{array}\right]\} \end{align}

There are many others.

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