Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the matrices and with some luck, choose representatives whose conjugacy class is large enough such that when I sum all the conjugacy class sizes I get the whole group.

  • 1
    $\begingroup$ $PGL(2,q)$ is just $GL(2,q)$ quotient the center. Thus conjugacy classes in $PGL(2,q)$ are just the equivalence classes of elements in $GL(2,q)$ given by the following relation: $g\equiv h$ if $g=h^kz$, where $g,h,k\in GL(2,q)$ and $z$ is central. Now central elements are multiples of the identity, so all you're doing is taking a normal conjugacy class in $GL(2,q)$, and maybe doing some scalar multiplication. In particular, Jordan normal forms mostly solve the problem. $\endgroup$ – user641 Mar 1 '12 at 18:53
  • $\begingroup$ What do you mean by $h^{k}$ in $g = h^{k}z$? $\endgroup$ – 387492 Mar 1 '12 at 21:41
  • $\begingroup$ The conjugate of $h$ by $k$, $h^k=k^{-1}hk$. $\endgroup$ – user641 Mar 1 '12 at 22:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.