As we have already said, investigating the properties of finite simple groups is often a good step for describing the properties of finite groups as a whole. This is true in particular for diameters. A consequence of Schreier’s lemma [Sch27], which we state as Lemma 5.2.1, is that the diameter of a finite group can essentially be bounded linearly in terms of the product of the diameters of its composition factors (see in fact Lemma 6.2.5, in the same spirit). When the group is the direct product of finite simple groups, the dependence is even nicer: its diameter can be bounded linearly by the maximum diameter of its factors, as shown in the author’s preprint [Don19a] based on previous results by Babai and Seress [BS92]
and Helfgott [Hel18] (see §5).
Our objective then becomes to estimate, as accurately as we are able to, the diameter of finite simple groups. By CFSG, we only need to do so for a handful of well-described families, for which we desire to give bounds depending on their size (or on the parameters that determine them, such asn andq). The sporadic groups in point (d) of Theorem 1.2.4 do not pose a problem at all: they are finitely many, albeit possibly very big (M, the largest one, approaches size 1054), so their diameter is just a constant that is even computable in principle. We have already seen in §1.1 that diam(Z/pZ) = p
2
, i.e. we have a linear dependence on|G| for Gfinite simple abelian. We are thus left with two cases to examine: alternating groups and groups of Lie type.
Conjecture 1.3.1 (Babai’s conjecture). Let G be a finite simple non-abelian group. Then, there is an absolute constantC >0 such that
diam(G)≤logC|G|.
From what we discussed in§1.1, this is essentially best possible: already for G= Alt(n), a lower bound is known with C= 2−εfor anyε >0 and |G|large enough. In fact, for anyε >0, given any finite groupGlarge enough with respect to ε and given a non-redundant setA of generators of G(meaning that there is no proper subset of A that still generatesG), we must have |A| ≤ log2|G| and then diam(Cay(G, A))≥log1−ε|G|. Therefore, even a bound witho(1) instead of C would be false for any infinite class of finite groups.
Babai’s conjecture was stated in the literature for the first time in 1988 by Babai and Seress [BS88, Conj. 1.7], and in its full generality it is still unsolved.
There have been however numerous weaker or partial results, which we are going to illustrate here.
As we showed, two infinite classes of finite non-abelian simple groups exist, the alternating groups and the groups of Lie type, and results often apply to only one of the two. Let us start with Alt(n). The oldest nontrivial bound appears in the same paper by Babai and Seress [BS88] in which Babai’s conjecture is first reported: their “modest first step”, as they called it, was to show that
diam(Sym(n)),diam(Alt(n))≤e(1+o(1))
√nlogn. (1.3.1) Soon after, they went on to show that the case of transitive permutation groups reduces to Alt(n), by claiming that for anyG≤Sym(n) transitive we have
diam(G) =eO(log3n)diam(Alt(m)) (1.3.2) where Alt(m) is the largest alternating group that is a composition factor of G (see [BS92, Thm. 1.4]). The proof however contains a bookkeeping mistake (as pointed out by Pyber, see [Hel18,§1]); the correct statement should be
diam(G) =eO(log2n)Y
i
diam(Alt(mi)) (1.3.3) with Q
imi ≤ n, as shown in [Hel18, Prop. 4.15] (known to Pyber). Setting aside the correctness of (1.3.2), results of this kind are in any case particularly significant for us, because combining them with (1.3.1) we obtain diameter bounds for all transitive groups; however, while (1.3.1) does not rely on CFSG (it has a purely combinatorial proof), both (1.3.2) and (1.3.3) follow from Cameron’s theorem which in turn, as we mentioned in§1.2, follows from CFSG: it will be the objective of§6 to try (and only partially succeed) to give a CFSG-free version of (1.3.3).
The next step, or giant leap, came at the hands of Helfgott and Seress [HS14]:
they proved that
diam(Sym(n)),diam(Alt(n))≤eO(log4nlog logn). (1.3.4) This quasipolynomial bound innis much closer to Babai’s conjecture than (1.3.1):
since|Alt(n)|= 12n!, a polylogarithmic bound in |G|as in Conjecture 1.3.1 corre-sponds to a polynomial bound in n. In particular, combining (1.3.4) and (1.3.3) (in fact they used the incorrect (1.3.2), but one can replace it with (1.3.3) without changing the end result, as stated in [Hel18, §1]), Helfgott and Seress settled a different conjecture stated in [BS88, Conj. 1.6], which claimed that the diameter of transitive groups is bounded quasipolynomially inn(a stronger polynomial con-jecture has also been formulated [KMS84]). The upper bound for diam(Alt(n)) in (1.3.4) is the best to date.
Let us turn now to groups of Lie type. Here, the advancements went mostly hand in hand with generalizations of the following theorem.
Theorem 1.3.2. Letpbe a prime, letG= SL2(Fp),PSL2(Fp), and letAbe a set of generators ofG. Then there exist absolute constantsδ >0 andk≥1 such that at least one of the following alternatives holds:
(a) |A3| ≥ |A|1+δ;
(b) (A∪A−1∪ {e})k=G.
The theorem above, in this particular form, is due to Helfgott [Hel08]. It describes the behaviour of a set of generators as being subject to an alternative:
any such set either has large growth or quickly fills the entire group. Notice that, as we pointed out in §1.1, “growth” is measured by the ratio |A|A|3|. Any theorem displaying the same kind of dichotomy is commonly referred to as a product theorem (as in [Bre14] [Raz14] [Hel18], to name a few instances). The importance of product theorems in the context of diameters is obvious: applying repeatedly Theorem 1.3.2 toA,A3,A9, etc... until we fall into case (b), the number of steps (a) that we can pass through at most is bounded by log log|G|−log log|A|
log(1+δ) , and the diameter is bounded byk+(log|G|)log 3/log(1+δ); hence, Babai’s conjecture holds for the family of simple groups PSL2(Fp) (pprime) [Hel08].
Helfgott’s result, which dates back to 2005, was the first of a series of in-creasingly general proofs of Babai’s conjecture for classes of finite simple groups of Lie type. First of all, Theorem 1.3.2 holds with A3 = G replacing case (b) [NP11], with|A3| = Ω(|A|1+201) replacing case (a) [RS18], and cannot hold with δ≥ 16(log27−1) [BRD15]. The theorem was generalized for PSL2(Fq) with any prime powerqby Dinai [Din11], and for PSL3(Fp) by Helfgott [Hel11]; moreover, case (a) was shown to hold for all sets of generators of PSLn(Fp) that are not too large [GH11]. Afterwards, a product theorem that holds for all finite simple groups of Lie type of bounded rank (or equivalently, for all such groups but where δ depends on the rank) was proved independently by Breuillard, Green and Tao [BGT11] and by Pyber and Szab´o [PS16]: Babai’s conjecture thus holds in this case as well.
There are however limitations to product theorems. In the sense of Theo-rem 1.3.2, a product theoTheo-rem cannot hold for groups of Lie type with no condition on the rank, as the counterexample in [PS16, Ex. 77] shows. The same is true for Alt(n), with counterexamples in [Spi12, §4] and [PPSS12, Thm. 17]: a proof of Babai’s conjecture for Alt(n) therefore cannot pass directly through a statement as strong as Theorem 1.3.2; there exists however a weaker version of a product theorem that does hold in the alternating group case, and thanks to which one can prove a diameter bound almost as strong as (1.3.4): the result is [Hel18, Thm. 1.4], and we will see more of it in§6. A diameter bound for finite simple groups of Lie type of bounded base field (and unbounded rank) also exists: for such groups G, Biswas and Yang [BY17] proved that the diameter is at mosteO(
√
log|G|(log log|G|)3), and the exponent 3 has been further reduced to 2 in [HMPQ19].
Existing proofs of the two main cases of Babai’s conjecture, G= Alt(n) and Gof Lie type, do not mingle much, as all the results cited above show. However, there are some deep-running similarities that pop their head out of the water here and there: for example, the counterexamples to strong product theorems are structurally analogous, as recognized for instance in [BGH+14, §1]; the authors of [BY17] acknowledge that their search for a matrix of small degree and close to the identity reminds of the search for a permutation of small support in [BS88].
The use itself of a weakened product theorem in [Hel18] is a deliberate effort in bringing the alternating and the Lie type case closer together, “towards a unified perspective”. There is however one nontrivial diameter bound that holds for all finite simple groups: Breuillard and Tointon [BT16] have proved that for every ε >0 there is a constant Cε such that
diam(G)≤max{|G|ε, Cε} (1.3.5) for all finite non-abelian simple groups G (the bound in [BT16, Cor. 1.4] is for Cayley graphs with symmetric generating sets, to be precise). The proof has also the remarkable characteristic of being CFSG-free: for Alt(n), the best CFSG-free bound is (1.3.1), which is stronger than (1.3.5) but not as general, and in any case much weaker than (1.3.4). A CFSG-free result much closer in strength to (1.3.4) (and in spirit to [Hel18]) would be Theorem 6.3.6, if Conjecture 6.3.4 were to be true (see§6 for details).
Another possible way to unify the treatment of the two cases, yet to bear fruits, could be through F1, the field with one element. There are ways to define such an object (which in any case is neither a field nor a one-element object:
an astounding feat, lying twice in the short string of symbols “F1”) and more importantly to define objects over it, in the sense of algebraic geometry: what is relevant to us is that generally one shows that “GLn(F1) = SLn(F1) = Sym(n)”, whatever that means (see for instance [Lor18,§2.1.2]).
Yet another way to close the gap between alternating and Lie type case, subject of recent research, could be to try and put into practice the following suggestion by Pyber, based on [BBS04]. One might be able to prove Babai’s conjecture for all finite simple non-abelian groups by performing three steps: 1) finding quickly an element of “support” n(1−ε) for someε >0; 2) using this first element, finding quickly a second element of smallest “support”; 3) using this second element, concluding the proof. In [BBS04], an element as in (1) is already inside our set of generators of Alt(n), and the other two steps follow; one would hope to do the same for groups of Lie type, although even defining what “support” means in this situation is not obvious. There has been some very recent progress on this front, due to Halasi [Hal20] and Eberhard and Jezernik [EJ20].