Problems of Connectivity between the Sylow Graph,the Prime Graph and the Non-Commuting Graph of a Group

Abstract

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.

Two bounds on the noncommuting graph

“…: recall from [1] that Γ G is defined by vertices x, y ∈ G − Z(G) = V joined by an edge xy ∈ E if x do not commute with y), there is neither weight nor orientation, so µ(Ω) = x∈Ω deg(x) and σ(∂Ω) = |∂Ω|. Important contributions on Γ G can be found in [1,8,12], but the reader may refer to [16] for a recent survey 1 . Γ G has interesting properties: it is always connected, of diameter 2 and hamiltonian (see [1,Propositions 2.1,2,2]); moreover the planar and the regular cases are classified by [1,Propositions 2.3,2.6].…”