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Find group elements of $SL(2,3)$

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0

$begingroup$

I am starting to learn about group theory so I am still really new in this field. So, I have the group $SL(2,3)$ with the two generators

$$
h = begin{pmatrix}
0 & a \
-(1/a) & 0
end{pmatrix}$$

and
$$ r = e^{itheta}/sqrt{ 2} begin{pmatrix}
e^{iphi} & e^{ipsi m} \
-e^{-ipsi m} & e^{-iphi}
end{pmatrix}$$

where $a$ and $m$ are reals. I need to find all 24 elements of the group, but I do not have a clear idea how to do it. I know that I have to find all possible multiplications, but I do not see how to obtain 24 different matrices. Is there a recipe to obtain these elements and to understand this problem better? Thanks in advance.

group-theory finite-groups linear-groups

share|cite|improve this question

edited 2 days ago

the_fox

2,90031538

asked 2 days ago

morsmors

134

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know anything about the constants $a,b,c,d,e,f$? For some choices of values the two matrices won't even be in the group $SL_2(Bbb{F}_3)$. For some other choices they will be there, but won¨t generate the whole group. So I suspect that you have left out something essential. Possibly accidentally. Anyway, in its present form the question doesn't seem to be answerable.
  $endgroup$
  – Jyrki Lahtonen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have available, e.g., a matrix description of $SL(2, 3)$?
  $endgroup$
  – Travis
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the answer, I already put the form of the matrices, I now for the fact that they form the total group, but I do not know how to multiply the matrices to generate the 24 elements.
  $endgroup$
  – mors
  2 days ago
  

  
  
  
  

add a comment | 

0

$begingroup$

I am starting to learn about group theory so I am still really new in this field. So, I have the group $SL(2,3)$ with the two generators

$$
h = begin{pmatrix}
0 & a \
-(1/a) & 0
end{pmatrix}$$

and
$$ r = e^{itheta}/sqrt{ 2} begin{pmatrix}
e^{iphi} & e^{ipsi m} \
-e^{-ipsi m} & e^{-iphi}
end{pmatrix}$$

where $a$ and $m$ are reals. I need to find all 24 elements of the group, but I do not have a clear idea how to do it. I know that I have to find all possible multiplications, but I do not see how to obtain 24 different matrices. Is there a recipe to obtain these elements and to understand this problem better? Thanks in advance.

group-theory finite-groups linear-groups

share|cite|improve this question

edited 2 days ago

the_fox

2,90031538

asked 2 days ago

morsmors

134

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know anything about the constants $a,b,c,d,e,f$? For some choices of values the two matrices won't even be in the group $SL_2(Bbb{F}_3)$. For some other choices they will be there, but won¨t generate the whole group. So I suspect that you have left out something essential. Possibly accidentally. Anyway, in its present form the question doesn't seem to be answerable.
  $endgroup$
  – Jyrki Lahtonen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have available, e.g., a matrix description of $SL(2, 3)$?
  $endgroup$
  – Travis
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the answer, I already put the form of the matrices, I now for the fact that they form the total group, but I do not know how to multiply the matrices to generate the 24 elements.
  $endgroup$
  – mors
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

I am starting to learn about group theory so I am still really new in this field. So, I have the group $SL(2,3)$ with the two generators

$$
h = begin{pmatrix}
0 & a \
-(1/a) & 0
end{pmatrix}$$

and
$$ r = e^{itheta}/sqrt{ 2} begin{pmatrix}
e^{iphi} & e^{ipsi m} \
-e^{-ipsi m} & e^{-iphi}
end{pmatrix}$$

where $a$ and $m$ are reals. I need to find all 24 elements of the group, but I do not have a clear idea how to do it. I know that I have to find all possible multiplications, but I do not see how to obtain 24 different matrices. Is there a recipe to obtain these elements and to understand this problem better? Thanks in advance.

group-theory finite-groups linear-groups

share|cite|improve this question

edited 2 days ago

the_fox

2,90031538

asked 2 days ago

morsmors

134

$endgroup$

share|cite|improve this question

edited 2 days ago

the_fox

2,90031538

asked 2 days ago

morsmors

134

share|cite|improve this question

edited 2 days ago

the_fox

2,90031538

asked 2 days ago

morsmors

134

edited 2 days ago

the_fox

2,90031538

edited 2 days ago

the_fox

2,90031538

asked 2 days ago

morsmors

134

asked 2 days ago

morsmors

134