The generating graph of a profinite group
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Archiv der Mathematik
The generating graph of a profinite group Andrea Lucchini
Abstract. Let G be a 2-generated group. The generating graph Γ(G) of G is the graph whose vertices are the elements of G and where two vertices g and h are adjacent if G = g, h. This definition can be extended to a 2-generated profinite group G, considering in this case topological generation. We prove that the set V (G) of non-isolated vertices of Γ(G) is closed in G and that, if G is prosoluble, the graph Δ(G) obtained from Γ(G) by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinite group G with the property that Δ(G) has 2ℵ0 connected components. This implies that the so called “swap conjecture” does not hold for finitely generated profinite groups. We also prove that if an element of V (G) has finite degree in the graph Γ(G), then G is finite. Mathematics Subject Classification. 20D60. Keywords. Generating graph, Profinite groups, Swap conjecture.
1. Introduction. Given a group G, the generating graph Γ(G) is the graph with vertex set G where two elements x and y are adjacent if and only if G = x, y. There could be many isolated vertices in this graph. All of the elements in the Frattini subgroup will be isolated vertices, but we can also find isolated vertices outside the Frattini subgroup (for example, the elements of the Klein subgroup are isolated vertices in Γ(Sym(4))). Several strong structural results about Γ(G) are known in the case where G is simple, and this reflects the rich group theoretic structure of these groups. For example, if G is a nonabelian simple group, then the only isolated vertex of Γ(G) is the identity and the graph Δ(G) obtained by removing the isolated vertex is connected with diameter two and, if G is sufficiently large, admits a Hamiltonian cycle. In [5], it is proved that Δ(G) is connected, with diameter at most 3, if G is a finite soluble group.
A. Lucchini
Arch. Math.
Clearly the definitions of Γ(G) and Δ(G) can be extended to the case of a 2-generated profinite group G (in this case, we consider topological generation, i.e. we say that X generates G if the abstract subgroup generated by X is dense in G). We denote by V (G) the set of the vertices of Δ(G). We will prove in Section 2 (see Proposition 6) that V (G) is a closed subset of G. The profinite group G, being a compact topological group, can be seen as a probability space. If we denote by μ the normalized Haar measure on G, so that μ(G) = 1, we may consider the probability μ(V (G)) that a vertex of the generating graph Γ(G) is non-isolated. By [4, Remark 2.7(ii)], if G is a 2-generated pronilpotent group, then μ(V (G)) ≥ 6/π 2 . However it is possible to construct a 2-generated prosoluble group G with μ(V (G)) = 0. Indeed let H = C22 and let h1 , h2 , h3 be the non-trivial elements of H. For each odd prime p, write Np = Cp3 and define G = p odd Np H where for each odd prime p, the subgroup Np is H-stable and for (np,1 , np,2 , np,3 ) ∈ Np and hj ∈ H, ⎧ −1 −1 ⎨ (np,1 ,
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