[GAP Forum] Presentation for L(6,2)

[Dmitrii Pasechnik]

Dear Lenny,
"maximal parabolic" presentations for all
classical groups were given by Chevalley and Tits
(my attribution is from memory, so can be a bit off)

In the case of L(n,2), such
a presentation is in terms of generators of the form Eij, i<>j, where
is  Eij a matrix with 1 at the entry (i,j) and on the main diagonal,
0 elsewhere.
More precisely, you only need to take Eij with i<j and j=i+i.
(The subgroups generated by all Eij except one Epq, q=p-1, are
called maximal parabolics)

Write down presentations for each L(3,2) that sits in a 3x3-submatrix on the
main diagonal; take the resulting relations, the relations that Eij commutes
with Epq (for all pairs ij, pq where this happens), the relations
defining the upper triangular subgroup B=<Eij|i<j>,
and the relations arising from describing the action of each Eij, j=i-1
on B (i.e. of the form Eij Epq Eij=X(i,p,q), X(i,p,q) an element
of B). Finally, add relations Eij^2=1.
The result is a (faithful) presentation  for L(n,2).

Hope this helps,
Dmitrii



On Sat, Nov 14, 2009 at 03:33:12AM +0800, lenny wrote:
> Dear Forum:
>
> The Atlas has presentations for L(5,2) and L(7,2),
> but none for L(4,2) or L(6,2). Gap can fing one for L(4,2),
> but my computer runs out of memmory when trying to find onw for L(6,2).
> Is one known?
>
> Lenny Chastkofsky
>
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