[GAP Forum] group epimorphisms

[Erik Postma]

2009/5/5 Gabriel Bartolini <gabar at mai.liu.se>:
> Hi,
>
> I've been searching around but haven't found if it is possible to list all
> groups of order N generated by elements of orders m1 to mr, for example;
> all groups of order 32 generated by two elements x1, x2 of order 8 such that
> order(x1*x2)=2.
>
> Any suggestions would be appreciated.
>
> Regards,
> Gabriel

Hi Gabriel,

I can think of two approaches, one of which won't work for this
example and one of which will.

The first approach would be to form the group with presentation <x1,
x2 | x1^8 = x2^8 = (x1*x2)^2 = 1>, and see if it's finite. In this
case it isn't, so we'd continue with the next approach; if it is, say
of size N, you can see if it's possible to find all its normal
subgroups of order N/32. Then mod those out and see in which cases the
images of x1 and x2 still have the properties that you are looking for
(instead of having an order that divides 8, for example).

For the second approach, first find all groups of order 32 with the
command AllSmallGroups(32). Then for each group G, find the conjugacy
classes C1, C2, ..., of elements of order 8. Now find all
representative pairs of elements from those conjugacy classes: for
each class, take one representative element g1 as the first item in
the pair; compute the stabilizer S of g1 in the automorphism group of
G; then form a pair [g1, g2] for representatives g2 from all S-orbits
of C1, C2, ... . Finally, for each such pair [g1, g2], check the order
of g1 * g2 and whether g1 and g2 generate all of G.

Of course, once you've found a pair that works, you don't need to
check any of the other pairs for that group.

HTH,

Erik Postma.