[GAP Forum] An overgroup of Sylow in PSL( 2, 7^3 )

[Thomas Breuer]

Dear GAP Forum,

Anvita had asked for clarification about an unexpected result
of a GAP computation.

I think the problem arises from the documentation of 'IntermediateSubgroup'.
The GAP Reference Manual states:

  ‣ IntermediateGroup( G, U ) ─────────────────────────────────────── function
  
  This  routine tries to find a subgroup E of G, such that G > E > U holds. If
  U  is  maximal  in  G,  the  function  returns fail. This is done by finding
  minimal blocks for the operation of G on the right cosets of U.
  
The manual does not tell us what the function returns if the given subgroup
is *not* maximal.

It *should* say that an intermediate group is returned if the search for it
was successful,
and that 'fail' is returned if the given subgroup is not maximal
but the function gave up at some point.

(In Anvita's example, the function gives up because the index of the
given subgroup exceeds a prescribed bound of 10^6.)

All the best,
Thomas


On Sun, Feb 07, 2021 at 02:56:10PM +0700, Anvita wrote:
> Dear Forum,
> 
> I need to construct a proper subgroup of PSL( 2, 7^3 ) that contains a
> Sylow 2-subgroup.
> According to the manual, IntermediateGroup returns `fail' if the subgroup
> is maximal.
> But the Sylow 2-subgroup is not maximal in this case.
> Thank you in advance for any clarification.
> 
> Anvita
> 
> =====
> gap> L := PSL(2,7^3);
> <permutation group of size 20176632 with 2 generators>
> gap>
> gap> S := SylowSubgroup (L,2);
> <permutation group of size 8 with 3 generators>
> gap>
> gap> U := IntermediateGroup(L,S);
> fail
> =====
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