> < ^ Date: Fri, 23 May 1997 10:31:54 +1000
^ From: Burkhard.Hofling@maths.anu.edu.au <hofling@maths.anu.edu.au >
> < ^ Subject: Re: NormalSubgroups

Dear Forum,

in his mail of Thu, 22 May 97, David Joyner asks:

A simple question: I have a fairly large group (size=10712468422656000)
which I want to understand a little better. (I have a wild guess that G
might be the semidirect product of A16 and C2^10, with C2^10 normal.)
Unfortunately, NormalSubgroups doesn't seem to work on it:

```gap> G:=Group(g*g^-1,g*g^-1);
Group( ( 2,23,20,17, 8,10)( 3,24,30,28,16, 9)( 4,21,18, 7)( 5,22,19, 6)
(11,12,25,26)(13,14,27,15), ( 1, 5,18,29,27,25)( 3, 6,16,28,14,24)
( 4,19,30,23)( 7,15,12,10)( 8,11,13, 9)(20,26,22,21) )
gap> L:=NormalSubgroups(G);
Error, Function: <function> must be a function at
return Z( p ^ d ) ^ i ... in
fun( arg[i] ) called from
List( [ 0 .. d - 1 ], function ( i ) ... end ) called from
GF( p ) called from
RationalClassesElementaryAbelianSubgroup( N1, S1 ) called from
RationalClassesPElements( G, p, Sum( rationalClasses, Size ) ) called
from
...
```

I suppose the problem occurs instantly after you call the function=
NormalSubgroups. If this is the case, then the problem is probably that you=
have re-defined the internal function Z, which is used in GAP to denote=
finite field elements, e.g. by defining Z := Centre (G) or similar. Since=
this is an internal function, the only way to restore it is to restart GAP.

Note that you shouldn't use variable names which start with capital letters,=
since they are reserved for GAP's internal and library functions. (Since=
this restriction is very counter-intuitive and I tend to use one-letter=
uppercase variable names myself, I have put the following lines into my=
gap.rc file, which is read automatically when GAP starts up:

InternalZ := Z;
InternalE := E; # E is the other one-letter internal function

so that I can restore Z by typing Z := InternalZ without restarting).

Anyhow, the function 'NormalSubgroups' is probably much too slow to give you=
a result within reasonable time (such brute-force attempts seldomly work=
for large groups). Consider the following instead:

If your conjecture were true, then n=C2^10 is contained in the core of a=
Sylow 2-subgroup P of G, and if G/N is isomorphic with A16, it would be a=
simple nonabelian group and N is the core of P.

However,=20

```gap> N := PCore(G,2);;
gap> Size (N);
16384
gap> IsElementaryAbelian (N);
true
```

yields that N is elementary abelian of order 2^14.=20

Hope this helps.

Burkhard.

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