Results 1 to 5 of 5

Math Help - Psl(2, 9)

  1. #1
    Junior Member
    Joined
    Jul 2008
    Posts
    54

    Psl(2, 9)

    Can you explain me how to show that the PSL(2, 9) is isomorphic to Alt(6)?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by dimuk View Post
    Can you explain me how to show that the PSL(2, 9) is isomorphic to Alt(6)?
    clearly we have \mathbb{F}_9=\{m+ni: \ m,n \in \mathbb{F}_3 \}, where i^2 = -1. now consider these two elements of \text{SL}(2,9):

    A=\begin{pmatrix} 1 & 1+i \\ 0 & 1 \end{pmatrix}, \ \ \ B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. then a simple calculation shows that A^3=I_2, \ B^2=(AB)^5=-I_2. let

    a=AZ(\text{SL}(2,9)), \ b=BZ(\text{SL}(2,9)) be the elements of G=\text{PSL}(2,9) corresponding to A and B. then:

    a^3=b^2=(ab)^5=1_G. let H=<a,b> \simeq A_5. (since H and A_5 have the same presentation.) so [G:H]=6.

    now let X=\{a_jH: \ j=1, \ ... \ , 6 \} be the set of left cosets of H. define f:G \rightarrow S_X by f(g)=\sigma_g, \ \forall g \in G,

    where \sigma_g(a_jH)=ga_jH, \ j=1, \ ... \ , 6. clearly \sigma_g \in S_X and f is a non-trivial homomorphism. since G is simple,

    we must have \ker f = \{1 \}, i.e. f is an embedding. so G is isomorphic to a subgroup of S_X \simeq S_6. since |G|=360

    and A_6 is the only subgroup of S_6 which has order 360, we must have G \simeq A_6. \ \ \ \square
    Last edited by NonCommAlg; July 4th 2008 at 03:16 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2008
    Posts
    54

    Psl(2,9)

    Thank you. I got the idea.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by NonCommAlg View Post
    obviously the number of Sylow 5-subgroups of G is 6.
    What is the problem with having 36 Sylow 5-subgroups?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by ThePerfectHacker View Post
    What is the problem with having 36 Sylow 5-subgroups?
    that proof wasn't complete. see the edited version!
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum