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In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of discovery and confirmation. Specifically, I am referring to the question (posted 11 years ago) Groups that do not exist, which asked whether there were finite simple groups conjectured at some point that turned out not to exist.

To build upon that discussion, I am looking for a very specific kind of artifact from the history of group theory: explicit character tables for such conjectured but non-existent finite simple groups, if they ever were constructed. My understanding is that character tables were computed for many groups as part of the classification effort.

Question: Could anyone provide references to or copies of explicit (complete) character tables developed for finite simple groups that were later shown not to exist?

If possible, please elucidate the reasoning behind the exclusion.



Regarding Feit-Thompson Conjecture [FT62] mentioned in Dave Benson's answer, which states that for any two distinct odd primes $p$ and $q$, the integer $\frac{p^q - 1}{p - 1}$ never divides $\frac{q^p - 1}{q - 1}$, I am unsure to what extent this has been verified. Below is a (parallelized) Python script that confirms the conjecture holds for $p, q < 50000$ within 35 hours. It also identifies the pair $(p, q) = (17,3313)$, discovered in [S71], as noted in JoshuaZ's comment, for the sole non-coprime example in this range. The range could be significantly improved by the method in [S71].

Bonus question: Is there any other pair $(p,q)$ of this kind?

More recent checkings can be found in [G04] at B25 on page 125:

enter image description here

The last ckecking mentioned above is [DK04]. See also [M09].

References

[FT62] Feit, Walter; Thompson, John G. A solvability criterion for finite groups and some consequences. Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 968--970.
[S71] Stephens, N. M. On the Feit-Thompson conjecture. Math. Comp. 25 (1971), 625.
[DK04] Dilcher, Karl; Knauer, Joshua. On a conjecture of Feit and Thompson. High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, 169--178, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004.
[G04] Guy, Richard K. Unsolved problems in number theory. Third edition. Problem Books in Mathematics. Springer-Verlag, New York, 2004. xviii+437 pp.
[M09] Motose, Kaoru. Notes on the Feit-Thompson conjecture. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 2, 16--17.


Computation

    ...$time python3 check_primes_parallel.py
Pair not relatively prime: p=17, q=3313, gcd=112643

real    2066m4.640s
user    13490m17.916s
sys 1m1.613s

Code

# check_primes_parallel.py
from sympy import primerange, gcd
from multiprocessing import Pool
import itertools

def check_relative_prime_and_divisibility(pair):
    p, q = pair
    num1 = (pow(p, q) - 1) // (p - 1)
    num2 = (pow(q, p) - 1) // (q - 1)
    
    d = gcd(num1, num2)
    if d == 1:
        return None
    
    if num1 % num2 == 0:
        return f"Found divisible pair: p={p}, q={q}"
    else:
        return f"Pair not relatively prime: p={p}, q={q}, gcd={d}"

def find_prime_pairs(max_prime, num_cpus=8):
    # Generate a list of odd primes up to max_prime
    primes = list(primerange(3, max_prime))
    
    # Generate all unique combinations of two primes
    prime_pairs = list(itertools.combinations(primes, 2))

    # Create a pool of workers with the desired number of CPUs
    with Pool(processes=num_cpus) as pool:
        # Map the function over the prime pairs, distributed across the workers
        results = pool.map(check_relative_prime_and_divisibility, prime_pairs)
    
    # Filter out None results and print the rest
    for result in filter(None, results):
        print(result)

# Set a maximum prime number limit according to your computational power
max_prime = 50000 # You can adjust this value
find_prime_pairs(max_prime)
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    $\begingroup$ A related question would be whether it was ever conjectured that there existed more than one simple group with same character table, and it turned out there was only one. But I suspect the answer to this question is also no. $\endgroup$ Dec 9, 2023 at 19:27
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    $\begingroup$ @TimothyChow : We now know that for any give order, there are at most two no n-isomorphic simple groups of that order, and with that knowledge (and knowing all cases for which it occurs), it should be rather easy to exclude two non-isomorphic simple groups having the same character tables. $\endgroup$ Dec 9, 2023 at 20:15

1 Answer 1

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This is not really a proper answer, but it's way too long for a comment:

My understanding is that by the time a complete character table has been obtained, this is very strong evidence for the existence of a group. There is no example that I'm aware of, where a character table has been completely computed, and thereafter the group was shown not to exist.

In particular, the entire structure of the proof of the Feit-Thompson odd order theorem is to use group theory to get yourself into a bunch of very specific configurations, and in each one, a contradiction is obtained through character theory. A baby model for this is Suzuki's CA theorem, which served as a model for the proof, and which does the special case where all centralisers of non-trivial elements are abelian - in this case both the group theoretic part and the character theoretic part were much easier, and the contradiction at the end of the proof is very neat. In the Feit-Thompson paper, there is one case where the authors resort to generators and relations, but this is only because it was not known whether there exist distinct odd primes $p$ and $q$ such that $(p^q-1)/(p-1)$ divides $(q^p-1)/(q-1)$. I think this is still not known, but I'm not sure. There are papers of Kaoru Motose that study this. Maybe if there are such, you have your character table...

On the other hand, there are examples where the group theory almost worked, and a shift of prime was eventually used in order to obtain a contradiction. A great example of this, that I have studied extensively, is Ron Solomon's paper, "Finite groups with Sylow 2-subgroups of type .3". The 2-local structure of such a group with centraliser of involution an odd order extension of $\operatorname{Spin}_7(q)$ was elucidated in great detail, before switching to an odd prime to obtain a contradiction. It turns out that the 2-local structure is indeed consistent, and these non-existent finite groups have a perfectly good 2-completed classifying space, related to the Dwyer-Wilkerson finite loop space at the prime 2. The finite groups look like groups of Lie type, but with Weyl group the three dimensional $2$-adic reflection group $2 \times L_3(2)$.

In general, once you Bousfield-Kan $\mathbb{Z}$-complete the classifying space of a finite group, it fractures into pieces corresponding to the prime divisors of the group order. Each piece describes the fusion system of the group at that prime, but the completion process destroys most of the information about how the primes interact.

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    $\begingroup$ Regarding the conjecture about $\frac{p^q-1}{p-1}$ never dividing $\frac{q^p-1}{q-1}$, this is still open. We do know that the stronger version of the conjecture that the two are always relatively prime is false, with one counterexample being $(p,q) = (17,3313)$. I've been told verbally, but do not have a citation offhand that others have found other arguments for that case in Feit-Thompson which are simpler than their original argument, but are still more complicated than what one would need to do in that case if one had the conjecture. $\endgroup$
    – JoshuaZ
    Dec 9, 2023 at 13:54
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    $\begingroup$ I think there were examples where character tables of supposed simple groups were known before the group was known to exist. $\endgroup$ Dec 9, 2023 at 16:14
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    $\begingroup$ @GeoffRobinson E.g., the Monster, right? $\endgroup$ Dec 9, 2023 at 19:23
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    $\begingroup$ @TimothyChow : yes, and that example shows that a priori, it is far from straightforward to determine whether a simple group exists, give a supposed character table. $\endgroup$ Dec 9, 2023 at 20:18
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    $\begingroup$ On the Feit-Thompson conjecture: mathoverflow.net/questions/281043/feit-thompson-conjecture $\endgroup$ Dec 10, 2023 at 7:02

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