Notas àlgebra Alex

Grupo lineal de matrices 𝐺𝐿2 (𝔽3 )
𝑎
𝐺𝐿2 (𝔽3 ) = {𝐴 = [
𝑐
𝑏
] ; 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝔽3 = {0̅, 1̅, 2̅}, 𝑑𝑒𝑡𝐴 = 𝑎𝑑 − 𝑏𝑐 ≠ 0̅}
𝑑
Nota: |𝐺𝐿2 (𝔽3 )| = (32 − 1)(32 − 31 ) = 8 ∙ 6 = 48
Observaciòn, específicamente
0 1 0
𝐺𝐿2 (𝔽3 ) = {[
],[
1 0 2
1 0 2 0
],[
],[
0 1 0 2
2 0 1 0
],[
],[
0 1 1 2
1 0 2 0
],[
],[
1 1 1 1
2 0 2 0
],[
],[
2 2 1 2
⋯ ,[
1 0 2 0 1 0 2 0 1 1 2 1 1 1 1 2 2 1 2 2
],[
],[
],[
],[
],[
],[
],[
],[
],[
],⋯
0 1 0 1 0 2 0 2 0 1 0 1 0 2 0 2 0 2 0 2
⋯ ,[
1 1 1 1 1 2 2 1 2 1 2 1 2 2 2 2 1 2 1 0
],[
],[
],[
],[
],[
],[
],[
],[
],[
],⋯
1 2 1 0 0 1 1 1 1 0 2 0 0 1 1 0 1 1 1 1
⋯ ,[
0 1 1 1 1 0 2 0 0 1 1 0 2 1 2 2 1 2 2 2
],[
],[
],[
],[
],[
],[
],[
],[
],[
],⋯
1 2 2 1 2 1 2 1 2 2 2 2 2 2 2 1 2 2 1 2
⋯ ,[
0 1 1 1 2 2 2 0 2 0 2 0 1 2 1 0
],[
],[
],[
],[
],[
],[
],[
]}
1 2 2 0 2 0 1 1 2 2 1 2 1 0 1 2
2
],⋯
2
Grupo especial lineal 𝑆𝐿2 (𝔽3 )
𝑆𝐿2 (𝔽3 ) = {𝐴 = [
𝑎
𝑐
𝑏
] ; 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝔽3 = {0̅, 1̅, 2̅}, 𝑑𝑒𝑡𝐴 = 1}
𝑑
Observaciòn, específicamente
𝑆𝐿2 (𝔽3 )
1 0 0 1 0
= {[
],[
],[
0 1 2 0 2
2 2 1 0 1
[
],[
],[
1 0 1 1 2
1 0 1 0
],[
],[
0 2 1 1
0 0 1 1
],[
],[
1 2 2 2
2 2 0 1
],[
],[
1 0 2 0
2 1 1 2
],[
],[
2 2 0 2
1 2 1 2
],[
],[
1 0 2 0
0 2 0 1
],[
],[
2 1 2 1
2 1 1 1
],[
],[
2 1 2 0
2 0 2
0
],[
], [
0 1 0
1
2 2 1 2
],[
],[
1 1 1 2
2
]}
2
Obsrvaciòn: 𝜑: 𝐺𝐿2 (𝔽3 ) → 𝔽∗3 = {1̅, 2̅} dada por 𝜑(𝐴) = det(𝐴) es un homomorfismo de
grupos ya que 𝜑(𝐴𝐵) = det(𝐴𝐵) = det(𝐴) det(𝐵)
Además 𝐾𝑒𝑟(𝜑) = {𝐴: det(𝐴) = 1} = 𝑆𝐿2 (𝔽3 ) luego
𝐺𝐿2 (𝔽3 )
≅ 𝔽∗3
𝑆𝐿2 (𝔽3 )
1
],
0