The invariably generating graph of the alternating and symmetric groups

Abstract

<jats:title>Abstract</jats:title><jats:p>Given a finite group <jats:italic>G</jats:italic>, the invariably generating graph of <jats:italic>G</jats:italic> is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of <jats:italic>G</jats:italic>, and two classes are adjacent if and only if they invariably generate <jats:italic>G</jats:italic>. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most 6.</jats:p>

Connected Components in the Invariably Generating Graph of a Finite Group

“…In particular, if q 9 then Λ(S) has at most one isolated vertex. This is in contrast to the case of alternating or symmetric groups G, where Λ(G) can have an arbitrarily large number of isolated vertices (see [17,Theorem 1.2]).…”

“…Nevertheless, a combinatorial proof along the lines of [13, Theorem 3.1] might be feasible in order to show the following: if a finite simple group S is such that Λ + (S) is connected and not bipartite, then Λ + (S t ) is connected for every t β(S). At present, the connectedness of Λ + (S) for S simple is essentially only known for alternating groups [17], which somewhat limits the applications of such a result.…”

“…In [17], the following definition was given. For a finite group G, the invariably generating graph Λ(G) of G is the undirected graph whose vertices are the conjugacy classes of G different from {1}, and two vertices C and D are adjacent if C, D I = G. If G is not invariably 2-generated, Λ(G) is the empty graph.…”

“…Guralnick and Malle [22] and Kantor et al [26] independently proved that every finite simple group S is invariably 2-generated, so that Λ(S) is not the empty graph. In [17], the author studied the graph Λ(G), in the case where G is an alternating or symmetric group, and proved that Λ + (G) is connected with diameter at most 6 in these cases (with the exception of S 6 , which is not invariably 2-generated). At present, it is not known whether Λ + (S) is connected if S is nonabelian simple (see [17,Question 1.6]).…”

“…In [17], an analogous graph, denoted by Λ e (G), was defined. Its vertices are the nontrivial elements of G, and two vertices are adjacent if the corresponding classes invariably generate G. In [20], it was proved that if S is nonabelian simple, then either S belongs to three explicit families of examples, or the proportion of isolated vertices of Λ e (S) tends to zero as |S| → ∞.…”