finite groups: class constants relation











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It is stated in Jansen, finite groups:



$c_l C_{j,k}^l=c_j C_{k(-l)}^{-j}$



where $c_l$ is the number of elements in class $l$ and where a fronting negative sign denotes the inverse class which is same as the class for ambivalent classes only and $c_l=c_{-l}$ always regardless. I understand and can reconcile the expression in the special case for $l=1$ meaning the class of identity in which case $c_l=1$ and the class of inverse $-l$ is same as for $l$ in this case since class of identity is ambivalent. In this case the only nonzero nontrivial result is for $j=-k,c_j=c_k$ being the number of elements in the class $k$. But cannot derive the general case and cannot find it anywhere on internet search nor other group theory texts.



It is also given :



$c_j c_k=sum C_{j,k}^l c_l$ sum over classes.



And



$C_{j,k}^l=C_{(-j),(-k)}^{-l}$



both of which I can reconcile but cannot derive the general case of the first expression above from these.






group-theory







share|cite|improve this question


























    up vote
    0
    down vote

    favorite












    It is stated in Jansen, finite groups:



    $c_l C_{j,k}^l=c_j C_{k(-l)}^{-j}$



    where $c_l$ is the number of elements in class $l$ and where a fronting negative sign denotes the inverse class which is same as the class for ambivalent classes only and $c_l=c_{-l}$ always regardless. I understand and can reconcile the expression in the special case for $l=1$ meaning the class of identity in which case $c_l=1$ and the class of inverse $-l$ is same as for $l$ in this case since class of identity is ambivalent. In this case the only nonzero nontrivial result is for $j=-k,c_j=c_k$ being the number of elements in the class $k$. But cannot derive the general case and cannot find it anywhere on internet search nor other group theory texts.



    It is also given :



    $c_j c_k=sum C_{j,k}^l c_l$ sum over classes.



    And



    $C_{j,k}^l=C_{(-j),(-k)}^{-l}$



    both of which I can reconcile but cannot derive the general case of the first expression above from these.






    group-theory







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      It is stated in Jansen, finite groups:



      $c_l C_{j,k}^l=c_j C_{k(-l)}^{-j}$



      where $c_l$ is the number of elements in class $l$ and where a fronting negative sign denotes the inverse class which is same as the class for ambivalent classes only and $c_l=c_{-l}$ always regardless. I understand and can reconcile the expression in the special case for $l=1$ meaning the class of identity in which case $c_l=1$ and the class of inverse $-l$ is same as for $l$ in this case since class of identity is ambivalent. In this case the only nonzero nontrivial result is for $j=-k,c_j=c_k$ being the number of elements in the class $k$. But cannot derive the general case and cannot find it anywhere on internet search nor other group theory texts.



      It is also given :



      $c_j c_k=sum C_{j,k}^l c_l$ sum over classes.



      And



      $C_{j,k}^l=C_{(-j),(-k)}^{-l}$



      both of which I can reconcile but cannot derive the general case of the first expression above from these.






      group-theory







      share|cite|improve this question













      It is stated in Jansen, finite groups:



      $c_l C_{j,k}^l=c_j C_{k(-l)}^{-j}$



      where $c_l$ is the number of elements in class $l$ and where a fronting negative sign denotes the inverse class which is same as the class for ambivalent classes only and $c_l=c_{-l}$ always regardless. I understand and can reconcile the expression in the special case for $l=1$ meaning the class of identity in which case $c_l=1$ and the class of inverse $-l$ is same as for $l$ in this case since class of identity is ambivalent. In this case the only nonzero nontrivial result is for $j=-k,c_j=c_k$ being the number of elements in the class $k$. But cannot derive the general case and cannot find it anywhere on internet search nor other group theory texts.



      It is also given :



      $c_j c_k=sum C_{j,k}^l c_l$ sum over classes.



      And



      $C_{j,k}^l=C_{(-j),(-k)}^{-l}$



      both of which I can reconcile but cannot derive the general case of the first expression above from these.






      group-theory




      group-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 22 at 10:55









      user158293

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