[GAP Forum] Low index normal subgroups

[Alexander Hulpke]

Dear Gap Forum,

Primoz Moravec asked:

> what is the best way to compute normal subgroups of low index in a given
> finitely presented group?

As Marco costatini already pointed out, the methods will be quite different
if you know that the group is finite (``finite'' here probably means: 
order <10^15 ).

If the group is finite, the best approach is to convert to a permutation
group or pc group and use for example `NormalSubgroups'.


If the group is potentially infinite or very large, this might be impossible
or infeasible. In this case I would do the following (assuming again that
your ``small index'' is really small, say up to n=1000:

Using the library of small groups of order <=n (these are the potential
factor groups), determine
- For all solvable groups of order <= n the orders of chief factors. You can
  use these as input to the solvable quotient algorithm to obtain a large
  finite solvable quotient of your group which has all solvable factor
  groups as factors.

- For each nonsolvable group $G$ determine a series of subgroups U1>...>Uk
  such that the image of the action of G on the cosets of Ui is nonsolvable
  and that the action on the cosets of Uk is faithful (often only one
  subgroup suffices, but for groups such as 2.A6 you might want to take a
  subgroup of inder 6 for U1). Then using `LowIndexSubgroups' determine all
  subgroups of the finitely presented group of index $[G:U1]$. Check whether
  the action on the cosets is permutation isomorphic to the action of G on
  the cosets of U1 (for example using `TransitiveIdentification' if the
  index is <=30).
  If U\le F is such a subgroup with [F:U]=[G:U1] determine subgroups of
  index [U1:U2] in U by the low index algorithm. Again compare the types of
  factor groups.
  Iterate until Uk (if not ruled out).

Most nonsolvable groups of large order have a small-degree permutation
representation or a factror group with a low-degree permutation
representation. Thus the second step of this program is probably
comparatively little work.

Best regards,

  Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke