> < ^ Date: Fri, 14 May 1999 18:12:01 +0200 (CEST)
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
< ^ Subject: Re: Group rings in GAP4

Dear GAP Forum,

Alexander B. Konovalov asked three questions.

1. It was a pleasant surprise for me to discover that in GAP4 group rings
are built-in
without any additional share packages. But I still did nit managed to find
description
of "GroupRing" function in the manual. I hope that somebody may tell me
where it
lies.

The manual of the current beta release GAP 4B5 is still rather incomplete.
The release version will contain a chapter on magma rings.

2. I have tried to compute the group ring of the dihedral group of order 8.
As you can
see below, the number of its generators depends on whether the group is
described
as pc-group or permutation group. Whether it is right ?

```gap> G:=Group((1,2,3,4),(2,4));
Group([ (1,2,3,4), (2,4) ])
gap> G1:=DihedralGroup(8);
<pc group of size 8 with 3 generators>
gap> F:=GF(2);
GF(2)
gap> FG:=GroupRing(F,G);
<algebra-with-one over GF(2), with 2 generators>  # ??????????
gap> FG1:=GroupRing(F,G1);
<algebra-with-one over GF(2), with 4 generators> # ??????????
```

The number of (group) generators of the group clearly may depend on the
representation of the group.
The number of (algebra-with-one) generators of the group may be equal
to the number of group generators if the group is finite.
In the release version of GAP 4, this finiteness will be used
also in the case of pc-groups.
(But independent of the generators, the group rings shown above
are correct.)

3. How can I compute the unit group of the group ring ?
In both cases described above I get the same error:

gap> Units(FG);
Error no method found for operation INV with 1 argument
[...]

In the release version of GAP 4, this will be fixed, too.
But it should be mentioned that GAP 4 has no special methods (yet)
to deal with the unit groups of group rings;
in particular for group rings of p-groups over the prime field
in characteristic p, effective algorithms are known but not yet
implemented in GAP 4.
So currently unit group calculations in group rings are restricted
to very small group rings.

If you are interested in the current documentation and code for
magma rings,
feel free to ask me, and I will send them via e-mail.

Sorry for the inconveniences,
Thomas

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