Isomorphism between linear groups over field:F2

Statement

Let $n$ be a natural number. Then, we have isomorphisms between the following linear groups over field:F2:

$GL(n,2)\cong SL(n,2)\cong PGL(n,2)\cong PSL(n,2)$

where the isomorphisms arise from the usual subgroup, quotient and subquotient maps that relate these groups. In particular:

• The embedding of the subgroup $SL(n,2)$ in $GL(n,2)$ is an isomorphism, i.e., the subgroup is the whole group
• The quotient map from $GL(n,2)$ to $PGL(n,2)$ is an isomorphism, i.e., the kernel is trivial
• The embedding of $PSL(n,2)$ in $PGL(n,2)$ is an isomorphism, i.e., the subgroup is the whole group
• The quotient map from $SL(n,2)$ to $PSL(n,2)$ is an isomorphism, i.e., the kernel is trivial

Related facts

• Isomorphism between linear groups when degree power map is bijective