Suppose I have a homomorphism $f:G\rightarrow GL_n(\mathbb{F}_p)$ and I wish to form the semidirect product $E\rtimes_f G$ with $E$ being the elementary abelian group of order $p^n$.
The Semidirect product command in Magma takes arguments Semidirect(K,H,f):Grp, Grp, Map -> Grp
to produce the semidirect product $K\rtimes_f H$ as a group object. Here f
is a homorphism from H
into AutK
, where AutK
is the Automorphism group of K, formed (I assume) by AutomorphismGroup(K)
.
In Magma, I set GLnp := GL(n,p);
and make $f$ using f := hom< G -> GLnp | ... >;
and $E$ using E := AbelianGroup([p:i in [1..n]]);
. Then I make AutE := AutomorphismGroup(E);
. So now the problem is taking that f
and turning it into a homomorphism from G
to AutE
so that I can use it with SemidirectProduct.
The natural way I thought to do this was to construct an isomorphism g
from GLnp
to AutE
, then setting phi := g*f;
and forming SemidirectProduct(E,G,phi);
. I find the GrpAuto
class pretty difficult to work with, but managed to do this in a hack-y way when I had $n=3$ and $p=2$. However when I then tried to use the same method to form the group I really wanted, for which $n=10$ and $p=23$, the isomorphism from g
from GLnp
to AutE
blew up, I assume because Magma tries to take the cartesian product of the two sets and then form the map as a subset (or something like that). I tried a couple other ways of forming g
, and some worked at first, but then blew up when I used SemidirectProduct(E,G,phi);
.
My $G$ here only has order $2640$ and Magma handles $E$ just fine by itself, so it seems like a permutation group on the scale of $E \rtimes_f G$ should be be well within the computational limits of Magma. (I have made permutation groups around that size before!) The problem is finding a way around whatever Magma is doing that is costing so much memory.
So I have three questions about this.
Am I wrong? Is what I want to do simply outside the computational limits of Magma?
In general,
What is the best way to make a homomorphism into a
GrpAuto
object from aGrpMat
object?
and, most importantly of all,
How can I form $E\rtimes_f G$ in Magma given a homomorphism $f:G\rightarrow GL_n(\mathbb{F}_p)$, given as a map
f
fromGrpPC : G
toGL(n, GF(p))
?
of order 48, but you need to give all 5 PC-generators, and also (I am afraid) their conjugate relations on all of the other generators. Also your actions of p and q on the 10-dimensional module seem to satisfy p*q*p^-1=q^4, whereas the relation you gave, q^p=q^4, means p^-1*q*p=q^4.
– 2012-11-26