Isomorphism between linear groups over field:F2

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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

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