Except Isomorphisms

Theorem

We have the following isomorphisms:

$SL_{2} (q) ≅ Sp_{2} (q) ≅ SU_{2} (q)$ .

For $q$ odd, $PSL_{2} (q) ≅ Ω_{3} (q)$ .

$O_{2}^{\pm} (q) ≅ D_{2 (q \mp 1)}$ , $SO_{2}^{\pm} (q) ≅ Z_{q \mp 1} . (2, q)$ and $Ω_{2}^{\pm} (q) ≅ Z_{(q \mp 1) / (2, q - 1)}$ .

$Ω_{4}^{+} (q) ≅ SL_{2} (q) \circ SL_{2} (q) ≅ (2, q - 1) . (PSL_{2} (q) \times PSL_{2} (q))$ .

$Ω_{4}^{-} (q) ≅ PSL_{2} (q^{2})$ .

For $q$ odd, $PSp_{4} (q) ≅ Ω_{5} (q)$ .

$P Ω_{6}^{+} (q) ≅ PSL_{4} (q)$

$P Ω_{6}^{-} (q) ≅ PSU_{4} (q)$ .

$PSL_{2} (2) ≅ S_{3}$ .

$PSL_{2} (3) ≅ A_{4}$ .

$PSL_{2} (4) ≅ PSL_{2} (5) ≅ A_{5}$ .

$PSL_{2} (7) ≅ PSL_{3} (2)$ .

$PSL_{2} (9) ≅ A_{6}$ .

$PSL_{4} (2) ≅ A_{8}$ .

$PSU_{3} (2) ≅ 3^{2} . Q_{8}$ .

$PSU_{4} (2) ≅ PSp_{4} (3)$ .

$Sp_{4} (2) ≅ S_{6}$ .

Remark. It comes from except isomorphism.png.

yhyquest

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Except Isomorphisms

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