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Dear Gap Forum,

David Joyner asked:

> Does anyone know an easy way to construct a matrix group over a

> ring such as Z/p^nZ in GAP? For example, I've tried constructing the

> ring R of 2x2 matrices

> over Z/9Z, for example, using something like

[...]

> Eventually, I'd like to convert such a group into a permutation group

At the moment, GAP has problems with inverting matrices over rings, which

are not integral domains (the default inversion routines do a gaussian

elimination which assumes no zero divisors occur).

Ultimately, this is also the reason for `Units' failing (though I don't get

a result as you do, but an error message or even a crash).

We are working on correcting these shortcomings, but as the fixes involve

interaction with some of the more inner setup of matrix arithmetic, I would

expect a correction will not make it in a bugfix, but only in the

next release.

Coming back to your original problem:

To create a permutation representation, one can already use the following

(slightly convoluted) method:

r:=Integers mod 9; o:=One(r); m1:=[[1,2],[3,4]]*o; m2:=[[5,6],[7,8]]*o; g:=GroupWithGenerators([m1,m2]); # does not test invertability v:=AsSSortedList(r^2); act:=Action(g,v,OnRight);

(The problem of finding generators for the full GL(n,Z/9) still remains,

however.)

Apologies for these complications,

Alexander Hulpke

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