From:

~~~ Subject:

Jerry Bryan wrote in his e-mail message of 1995/01/02

I am still absorbing this article, which exceeds my current

knowledge of group theory. But at the risk of asking a dumb

question, doesn't the center of GE (and of G) in fact consist

of more than just the Superflip and the identity? Does it

not also include the Pons Asinorum and the composition of

the Pons Asinorum and the Superflip? Call the Pons Asinorum P

and the Superflip E. I think you are saying Z={I,E}. But

isn't the center {I,P,E,PE}, with subgroups {I,P}, {I,E},

{I,PE}, and {I}? These should all be central, and hence

also normal, I would think.

This is not a dump question. Clearly ``Pons Asinorum'' P looks very

regular, and it is not farfetched to think that it is central. But it is

not. Only one out of 332640 elements of GE (and of G) centralizes P.

That is to say that the index of the centralizer of P in GE has index

332640 in GE. Since all elements of GC commute with all elements of GE,

the index of the centralizer of P in G also has index 332640 in G.

Z is indeed the center of GE', GE, G, G', and GCE.

It is in fact not too difficult to find the centers.

Recall that GE consists of the elements ( c_1, c_2, ..., c_12; p ),

where c_1 + c_2 + ... c_12 = 0 (mod 2) and p in S_12.

Since we can permute the components c_i in any way

by conjugation with an appropriate element (0,0,...,0;p),

it follows that any central element must have c_1 = c_2 = ... = c_12.

Furthermore any central elemement must have a permutation p that is

central in S_12. So we see that we have exactely two elements in the

center of GE, namely (0,0,...,0;<id>) and (1,1,...,1;<id>).

An easy argument shows that this is also the center of GE'.

The same argumentation works for GC, but the element

(1,1,...,1;<id>) is not in GC, since 1 + 1 + ... + 1 <> 0 (mod 3)

(since we have 8 summands). So GC has trivial center.

Again an easy argument shows that this is also the center of GC'.

The center of the direct product GCE is of course the direct product of

the centers of GC and GE. So we see that the center of GCE is again Z.

And again an easy argument shows that this is also the center of G.

If you have more questions, please do ask. I have tried to make my

article selfcontained. I think the only result that I used without

proof is that S_8 and S_12 have only one proper normal subgroup.

The problem is that in order to keep the article reasonably short,

I had to be rather terse at several places.

Have a nice day.

Martin.

-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany