_{1}

^{*}

The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid CFSG or not. On the other hand, what happens for infinite groups? Since the status of knowledge of the non-commuting graph and of the prime graph is satisfactory, is it possible to find relations between these two graphs and the Sylow graph? In the present note we make the point of the situation and formulate the above questions in appropriate way.

There is a large literature which is devoted to study the ways in which one can associate a graph with a group, for the purpose of investigating the algebraic structure using properties of the associated graph (see [1-4]). Many authors studied the so-called commuting graph of a group G, whose vertices are the non-trivial elements of G and two vertices x, y are connected if xy = yx (see [5-8]). The connectivity of gives information of how far is G from being abelian, therefore the commuting graph of abelian groups is always connected. In particular, is a connected component of and this motivates us to focuse only on , which gives significant information on the connectivity of.

By duality, the non-commuting graph, associated to a group G, is the complement of, i.e., a graph with vertex set, where two distinct non-central elements x and y of G are joined by an edge if and only if xy ≠ yx. Obviously, we are always considering simple graphs through this paper, i.e., graphs with no loops or directed or repeated edges. The non-commuting graph of a non-abelian finite group has received much attention in the last years and [9-20] are some significant contributions. Moreover, there are recent results in [

Herzog and others [

If G is finite, then is obviously related to the prime graph of G, whose vertices are the primes dividing the order of G and two vertices p and q of are joined by an edge if and only if G has an element of order pq.

was introduced by Gruenberg and Kegel in an unpublished manuscript [

Theorem 1.1 (See [8, Proposition 9]). and have the same number of connected components in a finite group G.

Since has been largely studied in [19,38-47] and in further works of the same authors, Theorem 1.1 allows us to apply results on for investigating and viceversa.

In the infinite case, some variations can be discussed once appropriate restrictions are done.

For instance, if G is an infinite group either residually solvable or residually finite, then is complete if and only if G is abelian.

On the other hand, we know from [28-33] that, the prime graph of a finite solvable group H has at most two connected components and, using CFSG, Williams [

Moghaddamfar, Khosravi and other authors worked on the same line of investigation in [38,42-45,47] and they provide interesting characterizations of simple groups, only looking at their prime graph.

In the infinite case, a systematic treatment seems to be absent in literature. For instance, Herzog and others [

Theorem 1.2. (See [8, Theorems 10, 12]) If G is a periodic solvable group, then has at most two components, each of diameter at most 9. Moreover, if G is locally finite (not necessarily solvable), then has at most 6 components, each of diameter at most 19.

Roughly speaking, Theorem 1.2 and its converse would allow us to recognize an infinite group from its commuting graph. To the best of our knowledge, we haven’t found whether an infinite locally finite simple group can be recognized from information on its commuting graph or not. This is a missing point which we noted.

Problem 1.3. Recognition of an infinite locally finite simple group from its commuting graph and from its prime graph.

If G is a finite group, then we have seen that, and have relations. Some recent contributions of Iranmanesh and others [14,15,46] show that also and may be related.

Theorem 2.1. (See [15, Lemmas 2.1, 2.2, Theorem 2.5, Corollary 2.7]) Let G and H be two non-abelian finite groups and (resp.) denote the set of orders of maximal abelian subgroups of G (resp. H).

1) If, then.

2) If G is simple and is isomorphic to, then.

3) If and is isomorphic to

, then and.

A consequence of Theorem 2.1 is [15, Corollary 2.8], where it is shown that for all even integers n bigger than 4, a finite group G is isomorphic to, provided is isomorphic to (here is a simple group of Lie type in the usual notation of [

Conjecture 2.2. (See [49, Problem 16.1]) Let S be a non-abelian finite simple group and G a group such that is isomorphic to. Then S is isomorphic to G.

Iranmanesh and others [14,46] dealt with further generalizations of Theorem 2.1. There are in fact evidences which show that the following conjecture is true.

Conjecture 2.3. (See [46, Problem 1]) Let G and H be two non-abelian finite groups such that is isomorphic to. If and have the same prime divisors and, then

.

In virtue of all we have said up to now, we haven’t found information whether a variation of Theorem 2.1 holds or not in the infinite case. The references [50-53] show that there is a theory which we may use for this scope, even if the notion of Sylow subgroup should be examined carefully.

Problem 2.4. Given two infinite locally finite non-abelian groups, is it possible to get a variation of Theorem 2.1?

In the previous section, we have described only relations among the non-commuting graph, the commuting graph and the prime graph, both in finite and infinite case. A new graph, recently studied in [54-56] has motivated us to write the present note and there would be interesting perspectives of study in our opinion.

Let be the set of prime divisors of the orders of elements of. Given a prime in, the automiser is the group. The Sylow graph of G, with set of vertices, is given by the following rules: Two vertices p, q in form an edge of if either q is in or p is in.

If G is finite, then coincides with the set of all prime divisors of and has been studied by Kazarin and others in [

Theorem 3.1. (See [55, Main Theorem]) is connected and its diameter is at most 5 whenever G is a finite almost simple group.

For solvable groups, the answer was known earlier as corollary of results in [54,57-60], where a different approach was adopted. [55, Main Theorem] influenced heavily [

Problem 3.2. Can we prove Theorem 3.1 without CFSG?

Another question has combinatorial interest. Lucido [

Problem 3.3. Is it possible to find relations between the connected components of and those of, when G is a finite group?

Once we move in the universe of all infinite groups, locally finite groups and periodic solvable groups seem to be natural contexts where we may generalize the notion of Sylow graph. Again [50-53] motivate this assertion. Of course, we cannot say that infinite locally finite groups satisfy Sylow Theorems, in order to carry out what we know from the finite case, but, results of Asar (see [

Problem 3.4. It is missing a theory of Sylow graphs for infinite groups, which agrees with that of the finite case. Infinite locally finite groups seem to be good candidates for a first attempt.

Now we come back to the finite case (and we will deal only with finite groups until the end) and note that the knowledge of the Sylow graph and of the properties of the normalizers of the Sylow subgroups is relevant for various reasons.

If G is a (finite) group in which every Sylow subgroup is self-normalizing, then [62,63] show that G is a p-group for some prime p. [62,63] are classical results and belong to a long standing line of research which investigates the structural properties of a group, once restrictions on its normalizers are given (see [61, Chapter 5] or [64-71]).

As noted in [

N was largely studied in [54-60], but the study of properties of normalisers appear also [62-72]. For instance, [63, Corollary 2] shows that the class of all p-groups is N-closed and [68, Theorem 2] shows that the same is true for the class of all nilpotent groups.

A further improvement is [64, Theorem 2], where it is considered the class of all p-nilpotent groups. In the class of all solvable groups, relations among N and classes of groups which are closed with respect to forming subgroups (briefly said S-closed), were investigated in [56- 60]. These researches make strong use of some techniques of the theory of formations of groups, which has been largely exploited in the last years by BallesterBolinches, Shemetkov and Skiba (terminology and notations which we are using can be found in fact in [61,67]). For instance, a significant notion is that of lattice-formation, which is quite technical to explain here (see [56-60, 67]), but has a fundamental role in many topics of formations of finite groups. Indeed, an immediate consequence of Theorem 3.1 is:

Theorem 3.5. (See [

Therefore allows us to decide whether a class of groups is N-closed or not. Many other interesting applications can be found in [55, Section 3]. We note that there are some interesting examples also in [55,56], which help to visualize the topic.

A final observation will have computational nature. The following condition is very easy to check empirically and, when it is true, the Sylow graph is connected.

Hypothesis. Assume that in a (finite) group G, if p is an odd prime in, then there exists another prime q in such that q < p and q divides

for each Sylow p-subgroup G_{p} of G.

Based on the previous Hypothesis, it has been written an algorithm with GAP [

This allows us to ask:

Problem 3.6. How can we improve numerical algorithms which confirm Theorem 3.1?

I thank the referees for comments and suggestions.