From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Regular polytopes
Date: 23 Nov 2000 16:37:27 GMT
Newsgroups: sci.math
Summary: [missing]

In article <B64193C9.203A%n_sotto@club-internet.fr>,
	Nicola Sottocornola <n_sotto@club-internet.fr> writes:
>Hello,
>I have some questions about some subgroups of O(4):
>
>1) do you know the order of the symmetry groups of the regular polytopes of
>R^4?

I am tempted to just say 'yes' to that.
You can find them listed for example in 'Regular Polytopes' by
HSM Coxeter. The polytoes are usually denoted by their Schl"afli symbol
{p,q,r}.

Polytope   Order of symmetry group.
{3,3,3}    120
{3,3,4}    384
{4,3,3}    384
{3,4,3}    1152
{3,3,5}    14400
{5,3,3}    14400

By the way, this refers to the full symmetry group, including reflections.
The rotation subgroup (determinant 1) has half of this order.

>2) for each of these groups, which is the maximal order of its elements?

That's a bit more tricky. But it is easy on a computer. The symmetry group
{p,q,r} is the group [p,q,r] defined by the Coxeter presentation
    <a,b,c,d | a^2,b^2,c^2,d^2,(ab)^p,(bc)^q,(cd)^r,(ac)^2,(ad)^2,(bd)^2 >
and so can be constructed explicitly.

Polytope   Orders of elements of full symmetry group
{3,3,3}    [1,2,3,4,5,6]
{3,3,4}    [1,2,3,4,6,8]
{4,3,3}    [1,2,3,4,6,8]
{3,4,3}    [1,2,3,4,6,8,12]
{3,3,5}    [1,2,3,4,5,6,10,12,15,20,30]
{5,3,3}    [1,2,3,4,5,6,10,12,15,20,30]


Derek Holt.
==============================================================================

From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Regular polytopes
Date: 25 Nov 2000 12:28:00 GMT
Newsgroups: sci.math

In article <B64470F2.2364%n_sotto@club-internet.fr>,
	Nicola Sottocornola <n_sotto@club-internet.fr> writes:
>Thank you Derek,
>where can I find these presentations? In the book of Coxeter?
>Nicola.
>
>> That's a bit more tricky. But it is easy on a computer. The symmetry group
>> {p,q,r} is the group [p,q,r] defined by the Coxeter presentation
>> <a,b,c,d | a^2,b^2,c^2,d^2,(ab)^p,(bc)^q,(cd)^r,(ac)^2,(ad)^2,(bd)^2 >
>> and so can be constructed explicitly.

You can find the presentations conveniently in another book:

H.S.M.Coxeter & W.O.J. Moser, Generators and Relations for Discrete
Groups, 4th edition, Springer-Verlag 1984.

I don't know whether that is the latest edition. I am afraid it is out of
print at present - I have been on the lookout for a copy to buy for the
past few years without any luck.

Derek Holt.