Number of prime factors of the order of a finite non-abelian simple group

Question

Hi everybody!

I am looking for results about how to bound from above the number of prime factors of the order of a non-abelian simple group $S$ in terms of, say, the index of a subgroup of $S$. I call $\omega(x)$ the number of prime factors of the integer $x$. For alternating groups everything is very simple, and we clearly have $\omega(|A_n|) \leq n$ (even better, but this is ok). Do you know results in this direction about the other simple groups? In particular I have two questions:

1) Let $S$ be a non-abelian simple group, and let $m(S)$ be the minimum index of a proper subgroup of $S$. Is it true that $\omega(|S|) \leq m(S)$?

2) Does there exist a sequence $\{S(n)\}_n$ of non-abelian simple groups such that $|S(n)|$ goes to infinity and $\{\omega(|S(n)|)\}_n$ is bounded?

Answer 1

To help searching: ω(|G|) = |π(G)|, and I see the latter usually.

A finite simple group G with |π(G)| = 1 must be cyclic of order p. By Burnside's p^aq^b theorem, if |π(G)|=2, then G is not simple.

The finite simple groups with |π(G)| = 3 were handled in several specific cases are handled by Brauer, Herzog, Klinger, Leon, Mason, Thompson, and Wales. In particular, it is now known any such group is one of the eight groups listed by Leon–Wales (1974), but this was not known as late as 1976. An important technique is to consider the list of minimal simple non-abelian groups as classified by Thompson in his N-groups papers. This narrowed the problem down to {2,3,p}-groups, and indeed p had only a few possibilities. Brauer, Leon, and Wales applied character theoretic techniques to classify such groups, and Klinger, Mason, and Thompson used local group-theoretic methods.

The finite simple groups with |π(G)| = 4 may have been classified by Cao. With the benefit of the classification, one has explicit order formulas for each finite simple group. Unfortunately, the prime factorizations of these formulas can be very difficult to understand. From the review of Bugeaud–Cao–Mignotte (2001), it appears that we probably have the complete list of groups, but that we may not have the proof of this in my lifetime (just due to silly things like Fermat and Mersenne primes).

In particular, I believe that it is not yet proven that there are only finitely many groups with |π(G)| = 4.

I would guess that effectively the answer to your 2nd question then is:

No there is no such sequence, but we cannot yet prove this due to number theory problems.

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Edit: Just to be fickle, let me mention another open problem in that intersection of finite simple groups and number theory. Solomon (2001) attributes this to Peter Neumann:

Are there infinitely many primes p such that |PSL(2, p)| is a product of six primes? (probably?)

My goodness there are a lot of them! 4721 up to |G| ≤ 10²⁰. Two of the primes have to be 2, one has to be 3, but the other three are sort of like "triplet primes", since they need to divide p−1, p, and p+1. The analogy is a little loose, since we divide (p−1)(p+1) by 24. The twin prime conjecture is still open, and Solomon describes these problems as "difficult and irrelevant obstacles" for understanding finite simple groups.

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I include a selection of papers. I think you get a flavor for the results from these, but there are several I left out, just because the list was getting long.

• Brauer, Richard. On simple groups of order 5⋅3^a⋅2^b. Bull. Amer. Math. Soc. 74 (1968) 900–903. MR236255 DOI:10.1090/S0002-9904-1968-12073-7

• Herzog, Marcel. On finite simple groups of order divisible by three primes only. J. Algebra 10 (1968) 383–388. MR233881 DOI:10.1016/0021-8693(68)90088-4 _{(See also: MR235037 MR249513 )}

• Thompson, John G. Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Amer. Math. Soc. 74 (1968) 383–437. MR230809 DOI:http://dx.doi.org/10.1090/S0002-9904-1968-11953-6

• Leon, Jeffrey S.; Wales, David B. Simple groups of order 2^a3^bp^c with cyclic Sylow p-groups. J. Algebra 29 (1974), 246–254. MR338153 DOI:10.1016/0021-8693(74)90098-2 _{(See also: MR265452 MR268270 MR281792 MR286882 MR308254 MR335623 )}

• Klinger, Kenneth. Finite groups of order 2^a3^b13^c. J. Algebra 41 (1976), no. 2, 303–326. MR414689 DOI:10.1016/0021-8693(76)90185-X

• Mason, Geoffrey. A characterization of SL(3, 3). I. J. Algebra 38 (1976), no. 1, 45–74. MR407129 DOI:10.1016/0021-8693(76)90243-X

• Cao, Zhen Fu. The simple groups of order 2^α₁3^α₂5^α₃7^α₄p^α₅. Chinese Ann. Math. Ser. A 16 (1995), no. 2, 244–250. MR1341938

• Solomon, Ronald. A brief history of the classification of the finite simple groups. Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 315–352. MR1824893 DOI:10.1090/S0273-0979-01-00909-0

• Bugeaud, Yann; Cao, Zhenfu; Mignotte, Maurice. On simple K₄-groups. J. Algebra 241 (2001), no. 2, 658–668. MR1843317 DOI:10.1006/jabr.2000.8742

• Zhang, Liangcai; Chen, Guiyun; Chen, Shunmin; Liu, Xuefeng. Notes on finite simple groups whose orders have three or four prime divisors. J. Algebra Appl. 8 (2009), no. 3, 389–399. MR2535997 DOI:10.1142/S0219498809003382

Answer 2

Let $F(x) \in \mathbf{Z}[x]$ be a non-constant polynomial. It's a standard fact in Sieving theory that there are infinitely many primes $p$ such that $F(p)$ has at most $r$-prime factors for some $r$ depending on $F(x)$. Moreover, given $F(x)$ one may compute explicitly a suitable $r$. In particular, the groups $\mathrm{PSL}(\mathbf{F}_p)$ will have order divisible by at most (say) $50$ primes for infinitely many $p$. The remark alluded to in the previous answer regarding "difficult number theory problems" only concerns finding the optimal value of $r$, which does lead to twin-prime like problems.

You can find such theorems in Ch.X of Halberstam and Richert's book, with the caveat that the statements there will probably assume that the values of $F(n)$ for $n \in \mathbf{Z}$ are not all divisible by a fixed prime $p$ (which is not true of $x^3 - x$ because of $p = 2$), but the arguments can easily be adapted to account for this.