1 Introduction

Given a group G,  the generating graph \(\Gamma (G)\) is the graph with vertex set G where two elements x and y are adjacent if and only if \(G=\langle x,y \rangle .\) 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 \(\Gamma ({{\,\mathrm{Sym}\,}}(4))).\) Several strong structural results about \(\Gamma (G)\) 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 non-abelian simple group, then the only isolated vertex of \(\Gamma (G)\) is the identity and the graph \(\Delta (G)\) 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 \(\Delta (G)\) is connected, with diameter at most 3, if G is a finite soluble group.

Clearly the definitions of \(\Gamma (G)\) and \(\Delta (G)\) 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 \(\Delta (G)\). 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 \(\mu \) the normalized Haar measure on G,  so that \(\mu (G) = 1,\) we may consider the probability \(\mu (V(G))\) that a vertex of the generating graph \(\Gamma (G)\) is non-isolated. By [4, Remark 2.7(ii)], if G is a 2-generated pronilpotent group, then \(\mu (V(G))\ge 6/\pi ^2.\) However it is possible to construct a 2-generated prosoluble group G with \(\mu (V(G))=0.\) Indeed let \(H = C_2^2\) and let \(h_1\), \(h_2\), \(h_3\) be the non-trivial elements of H. For each odd prime p,  write \(N_p = C_{p}^3\) and define \( G = \left( \prod _{p \text { odd }}N_p \right) \rtimes H \) where for each odd prime p,  the subgroup \(N_p\) is H-stable and for \((n_{p,1},n_{p,2},n_{p,3}) \in N_p\) and \(h_j \in H,\)

$$\begin{aligned} (n_{p,1},n_{p,2},n_{p,3})^{h_j} = \left\{ \begin{array}{ll} (n_{p,1},n_{p,2}^{-1},n_{p,3}^{-1}) &{} \text {if } j=1, \\ (n_{p,1}^{-1},n_{p,2},n_{p,3}^{-1}) &{} \text {if } j=2, \\ (n_{p,1}^{-1},n_{p,2}^{-1},n_{p,3}) &{} \text {if } j=3. \end{array}\right. \end{aligned}$$

The vertex \(((n_{p,i})_{1\le i\le 3, p \text { odd} };h)\) is non-isolated in \(\Gamma (G)\) if and only if \(h=h_j\) for some \(1 \le j \le 3\) and \(n_{p,j} \ne 0\) for all p (see [4, Example 2.8] for more details). This implies

$$\begin{aligned} \mu (V(G))= \frac{3}{4} \prod _{p \text { odd}}\left( 1-\frac{1}{p}\right) =0. \end{aligned}$$

The neighborhood of a vertex g of the generating graph of G, denoted by \({\mathcal {N}}_G(g),\) is the set of vertices of \(\Gamma (G)\) adjacent to g. The degree of g, denoted by \(\delta _G(g),\) is the number of edges of \(\Gamma (G)\) incident with g. Since \(G=\langle g, x\rangle \) if and only if there is no open maximal subgroup of G containing g and x, it follows that

$$\begin{aligned} {\mathcal {N}}_G(g)=G\setminus \bigcup _{M\in {\mathcal {M}}_g} M, \end{aligned}$$

where \({\mathcal {M}}_g\) is the set of the open maximal subgroups of G containing g. In particular, \({\mathcal {N}}_G(g)\) is a closed subset of V(G). The surprising result is that if \(\delta _G(g)\) is finite for some \(g\in V(G),\) then G is finite. Indeed we have:

Theorem 1

Assume that G is an infinite 2-generated profinite group. Then the degree of any non-isolated vertex in the generating graph \(\Gamma (G)\) is infinite.

This is a consequence of the following result, concerning the generating graph of a finite group G.

Theorem 2

Let G be a 2-generated finite group. If \(g\in V(G),\) then \(\delta _G(g)\ge 2^{t-2},\) where t is the length of a chief series of G.

The results concerning the connectivity of \(\Delta (G)\) when G is a finite soluble group can be extended with standard arguments to the prosoluble case.

Theorem 3

If G is a 2-generated prosoluble group, then \(\Delta (G)\) is connected and \({{\,\mathrm{diam}\,}}(\Delta (G)) \le 3.\)

The bound \({{\,\mathrm{diam}\,}}(\Delta (G))\le 3\) given in Theorem 3 is best possible. In [5], a soluble 2-generated group G of order \(2^{10}\cdot 3^2\) with \({{\,\mathrm{diam}\,}}(\Delta (G))=3\) is constructed. However, in [5], it is proved that \({{\,\mathrm{diam}\,}}(\Delta (G))\le 2\) in some relevant cases. These results can be extended to the profinite case, with the same arguments used in the proof of Theorem 3. Suppose that a 2-generated prosoluble soluble group G has the property that \(|{{\,\mathrm{End}\,}}_G(V)|>2\) for every non-trivial irreducible G-module V which is G-isomorphic to a complemented chief factor of G. Then \({{\,\mathrm{diam}\,}}(\Delta (G))\le 2\). In particular, \({{\,\mathrm{diam}\,}}(\Delta (G))\le 2\) if the derived subgroup of G is pronilpotent or has odd order (as a supernatural number).

No example is known of a 2-generated finite group G for which \(\Delta (G)\) is disconnected, and it is an open problem whether or not \(\Delta (G)\) is connected when G is an arbitrary finite group. The situation is different in the profinite case.

Theorem 4

There exists a 2-generated profinite group G with the property that V(G) is the disjoint union of \(2^{\aleph _0}\) connected components. Moreover each connected component is dense in V(G).

The previous theorem implies that the “swap conjecture” is not satisfied by the 2-generated profinite groups. Recall that the swap conjecture concerns the connectivity of the graph \(\Sigma _d(G)\) in which the vertices are the ordered generating d-tuples and two vertices \((x_1,\dots ,x_d)\) and \((y_1,\dots ,y_d)\) are adjacent if and only if they differ only by one entry. Tennant and Turner [8] conjectured that the swap graph is connected for every group. Roman’kov [6] proved that the free metabelian group of rank 3 does not satisfy this conjecture but no counterexample is known in the class of finite groups. However, by Lemma 10 in Section 3, it follows from Theorem 4 that there exists a 2-generated profinite group with the property that the graph \(\Sigma _2(G)\) has \(2^{\aleph _0}\) connected components.

2 Some properties of \(\Gamma (G)\)

We begin this section by proving a criterion to decide when a vertex x of the generating graph \(\Gamma (G)\) of a profinite group G is non-isolated.

Lemma 5

Let G be a 2-generated profinite group. Then \(x\in V(G)\) if and only if \(xN \in V(G/N)\) for every open normal subgroup N of G.

Proof

Let \({\mathcal {N}}\) be the set of the open normal subgroups of G. It is clear that if \(x\in V(G),\) then \(xN\in V(G/N)\) for every \(N\in {\mathcal {N}}.\) Conversely, assume \(xN \in V(G/N)\) for every \(N\in {\mathcal {N}}.\) Given \(N\in {\mathcal {N}},\) let \(\Omega _N=\{y\in G \mid \langle x,y \rangle N=G\}.\) Notice that if \(y\in \Omega _N,\) then \(yN\subseteq \Omega _N\) and consequently \(\Omega _N\), being a union of cosets of the open subgroup N,  is a non-empty closed subset of G. If \(N_1,\dots ,N_t\in {\mathcal {N}},\) then \(\varnothing \ne \Omega _{N_1\cap \dots \cap N_t}\subseteq \Omega _{N_1}\cap \dots \cap \Omega _{N_t}.\) Since G is compact, \(\cap _{N\in {\mathcal {N}}}\Omega _N\ne \varnothing .\) Let \(y\in \cap _{N\in {\mathcal {N}}}\Omega _N.\) Since \(\langle x,y \rangle N=G\) for every \(N\in {\mathcal {N}},\) we have \(\langle x,y \rangle =G,\) and consequently \(x\in V(G).\)\(\square \)

Proposition 6

If G is a 2-generated profinite group, then V(G) is a closed subset of G.

Proof

We prove that \(G\setminus V(G)\) is an open subset of G. Let \(x\notin V(G).\) By Lemma 5, there exists \(N\in {\mathcal {N}}\) such that \(x\notin V(G/N)\). This means that \(\langle x, y \rangle N \ne G\) for every \(y\in G,\) and consequently \(\langle xn, y \rangle \ne G\) for every \(n\in N\) and \(y\in G.\) This implies \(xN \cap V(G)=\varnothing ,\) so xN is an open neighbourhood of x contained in \(G\setminus V(G).\)\(\square \)

Proof of Theorem 3

By [5, Theorem 1], for every N in the set \({\mathcal {N}}\) of the open normal subgroups of G,  the graph \(\Delta (G/N)\) is connected and \({{\,\mathrm{diam}\,}}(G/N)\le 3.\) Let a and b be two distinct elements of V(G). For every \(N\in {\mathcal {N}},\) let \({{\,\mathrm{dist}\,}}_N(a,b)\) be the distance in the graph \(\Delta (G/N)\) of the two vertices aN and bN. Let

$$\begin{aligned} t:= \max _{N\in {\mathcal {N}}}{{\,\mathrm{dist}\,}}_N(a,b). \end{aligned}$$

Clearly \(t\le 3.\) Set \({\mathcal {M}}=\{N\in {\mathcal {N}} \mid {{\,\mathrm{dist}\,}}_N(a,b)=t\}.\) If \(N\in {\mathcal {N}}\) and \(M\in {\mathcal {M}},\) then \(N\cap M\in {\mathcal {M}},\) so \(\cap _{M\in {\mathcal {M}}}M=1.\) For every \(M\in {\mathcal {M}},\) let

$$\begin{aligned} \Omega _M\!=\!\{(x_1,\dots ,x_t)\!\in \! G^t\!\mid \!\langle x_1, x_2\rangle M\!=\!\cdots \!=\langle x_{t-1}, x_t\rangle M\!=\!G, x_1M\!=\!aM, x_tM\!=\!bM\}. \end{aligned}$$

If \((x_1,\dots ,x_t)\in \Omega _M,\) then \((x_1,\dots ,x_t)M^t\subseteq \Omega _M\), so \(\Omega _M\) is a closed subset of \(G^t.\) If \(M_1,\dots ,M_u\in {\mathcal {M}},\) then \(\varnothing \ne \Omega _{M_1\cap \dots \cap M_u}\subseteq \Omega _{M_1}\cap \dots \cap \Omega _{M_u}.\) Since G is compact, \(\cap _{M\in {\mathcal {M}}}\Omega _M\ne \varnothing .\) Let \((x_1,\dots ,x_t)\in \cap _{M\in {\mathcal {M}}}\Omega _M.\) Since \(\langle x_1,x_2 \rangle M=\dots =\langle x_{t-1},x_t \rangle M =G\) for every \(M\in {\mathcal {M}},\) we have \(\langle x_1,x_2 \rangle =\dots =\langle x_{t-1},x_t \rangle =G.\) Moreover \(x_1\in \cap _{M\in {\mathcal {M}}}aM=\{a\}\) and \(x_t\in \cap _{M\in {\mathcal {M}}}bM=\{b\}.\) We conclude that \((x_1,\dots ,x_t)\) is a path in \(\Delta (G)\) joining the vertices \(a=x_1\) and \(b=x_t.\)\(\square \)

3 An example

Let \(G_p=({{\,\mathrm{SL}\,}}(2,2^p))^{\delta _p},\) where p is a prime and \(\delta _p\) is the largest positive integer with the property that the direct power \(({{\,\mathrm{SL}\,}}(2,2^p))^{\delta _p}\) can be generated by 2-elements. The graph \(\Delta (G_p)\) is connected for every prime p,  and, by [2, Theorem 1.3], there exists an increasing sequence \((p_n)_{n\in {\mathbb {N}}}\) of odd primes such that \({{\,\mathrm{diam}\,}}(\Delta (G_{p_n}))\ge 2^n\) for every \(n\in {\mathbb {N}}.\) Consider the cartesian product

$$\begin{aligned} G=\prod _{n\in {\mathbb {N}}}G_{p_n} \end{aligned}$$

with the product topology. Notice that G is a 2-generated profinite group. Moreover \(\Delta (G)\) is the infinite tensor product of the finite graphs \(\Delta (G_{p_n}),\)\(n\in {\mathbb {N}},\) and \(V(G)=\prod _{n\in {\mathbb {N}}}V(G_{p_n}).\) Indeed \(\langle (x_n)_{n\in {\mathbb {N}}}, (y_n)_{n\in {\mathbb {N}}}\rangle =G\) if and only if \(\langle x_n, y_n\rangle =G_{p_n}\) for every \(n\in {\mathbb {N}}.\) First we describe the connected components of the graph \(\Delta (G).\)

Lemma 7

Let \(x=(x_n)_{n\in {\mathbb {N}}}\in V(G)\) and let \(\Omega _x\) be the connected component of \(\Delta (G)\) containing x. Then \(y=(y_n)_{n\in {\mathbb {N}}}\) belongs to \(\Omega _x\) if and only if

$$\begin{aligned} \sup _{n\in {\mathbb {N}}}{{\,\mathrm{dist}\,}}_{\Delta (G_{p_n})}(x_n,y_n)<\infty . \end{aligned}$$

Proof

Assume that \(y=(y_n)_{n\in {\mathbb {N}}}\in \Omega _x\) and let \(m={{\,\mathrm{dist}\,}}_{\Delta (G)}(x,y).\) It follows that \({{\,\mathrm{dist}\,}}_{\Delta (G_{p_n})}(x_n,y_n)\le m\) for every \(n\in {\mathbb {N}}.\) Conversely assume \(y=(y_n)_{n\in {\mathbb {N}}}\in V(G)\) with \({{\,\mathrm{dist}\,}}_{\Delta (G_{p_n})}(x_n,y_n)\le m\) for every \(n\in {\mathbb {N}}.\) There is a path

$$\begin{aligned} x_n=x_{n,0},x_{n,1},\dots ,x_{n,\mu _n}=y_n, \end{aligned}$$

with \(\mu _n\le m,\) joining \(x_n\) and \(y_n\) in the graph \(\Delta (G_{p_n}).\) For \(0\le i\le m,\) set

$$\begin{aligned} \begin{aligned} {\tilde{x}}_{n,i}={\left\{ \begin{array}{ll}x_{n,i}&{}\text {if } i < \mu _n,\\ x_{n,\mu _n}&{}\text {if } i\ge \mu _n \text { and } m-\mu _n \text { is even,}\\ x_{n,\mu _n-1}x_{n,\mu _n}&{}\text {if } i\ge \mu _n \text { and } m-\mu _n \text { is odd.} \end{array}\right. } \end{aligned} \end{aligned}$$

Then

$$\begin{aligned} x={\tilde{x}}_0=({\tilde{x}}_{n,0})_{n\in {\mathbb {N}}},\ {\tilde{x}}_1=({\tilde{x}}_{n,1})_{n\in {\mathbb {N}}},\dots ,y={\tilde{x}}_m=({\tilde{x}}_{n,m})_{n\in {\mathbb {N}}} \end{aligned}$$

is a path joining x and y in the graph \(\Delta (G),\) so \(y\in \Omega _x.\)\(\square \)

Proposition 8

\(\Delta (G)\) has \(2^\aleph _0\) different connected components.

Proof

Fix \(x=(x_n)_{n\in {\mathbb {N}}}\in \Delta (G).\) Let \(\tau \) be a real number with \(\tau >1.\) Since

$$\begin{aligned} {{\,\mathrm{diam}\,}}(\Delta (G_{p_n}))\ge 2^n\ge 1+\lfloor n/\tau \rfloor \end{aligned}$$

for every \(n\in {\mathbb {N}},\) there exists \(y_{\tau ,n}\in G_{p_n}\) such that \({{\,\mathrm{dist}\,}}_{\Delta (G_{p_n})}(x_n,y_{\tau , n})=1+\lfloor n/\tau \rfloor .\) If \(\tau _2>\tau _1,\) then

$$\begin{aligned} {{\,\mathrm{dist}\,}}(y_{\tau _2,n},y_{\tau _1,n})\ge {{\,\mathrm{dist}\,}}(x_n,y_{\tau _1,n})-{{\,\mathrm{dist}\,}}(x_n,y_{\tau _2,n})=\lfloor n/\tau _1 \rfloor - \lfloor n/\tau _2 \rfloor \end{aligned}$$

tends to infinity with n,  so, by Lemma 7, \(\Omega _{y_{\tau _1}}\ne \Omega _{y_{\tau _2}}.\)\(\square \)

Proposition 9

Let \(\Omega _x\) be the connected component of \(\Delta (G)\) containing x. Then \(\Omega _x\) is a dense subset of V(G),  and consequently it is not a closed subset of V(G).

Proof

Let \(y=(y_n)_{n\in {\mathbb {N}}}\in V(G).\) A base of open neighbourhoods of y consists of the subsets \(\Delta _{y,m}=\{(z_n)_{n\in {\mathbb {N}}}\mid z_n=y_n \text { for every } m\le n\}\) with \(m\in {\mathbb {N}}.\) In particular, set \(z=(z_n)_{n\in {\mathbb {N}}},\) with \(z_n=y_n\) if \(n\le m,\)\(z_n=x_n\) otherwise. We have \(z \in \Delta _{y,m}.\) Moreover, since

$$\begin{aligned} \sup _{n\in {\mathbb {N}}}{{\,\mathrm{dist}\,}}_{\Delta (G_{p_n})}(x_n,z_n) = \sup _{n\le m}{{\,\mathrm{dist}\,}}_{\Delta (G_{p_n})}(x_n,y_n)<\infty , \end{aligned}$$

it follows from Lemma 7 that \(z\in \Omega _x.\) Hence \(z\in \Omega _x \cap \Delta _{y,m}.\)\(\square \)

Lemma 10

If \((x_1,y_1)\) and \((x_2,y_2)\) are in the same connected component of \(\Sigma _2(G),\) then \(x_1, y_1, x_2, y_2\) are in the same connected component of \(\Delta (G).\)

Proof

It suffices to prove that if

$$\begin{aligned} (\alpha _1,\beta _1),\dots ,(\alpha _u,\beta _u) \end{aligned}$$

is a path in \(\Sigma _2(G)\), then the vertices \(\alpha _1,\beta _1,\alpha _2,\beta _2,\dots , \alpha _u, \beta _u\) belong to the same connected component of the graph \(\Delta (G).\) We prove this claim by induction on u. The statement is clearly true when \(u=1.\) Assume \(u\ge 2.\) By induction, the vertices \(\alpha _2,\beta _2,\dots , \alpha _u, \beta _u\) belong to the same connected component of \(\Delta (G);\) so it is enough to show that \(\alpha _1,\beta _1,\alpha _2, \beta _2\) belong to the same connected component. Since \((\alpha _1,\beta _1)\) and \((\alpha _2,\beta _2)\) differ only by one entry, either \(\alpha _1=\alpha _2\) or \(\beta _1=\beta _2.\) The graph \(\Delta (G)\) contains the path \(\beta _1,\alpha _1=\alpha _2,\beta _2\) in the first case and the path \(\alpha _1,\beta _1=\beta _2,\alpha _2\) in the second case. \(\square \)

4 Degrees in the generating graph

Before proving Theorem 1, we briefly recall some necessary definitions and results. Given a subset X of a finite group G,  we will denote by \(d_X(G)\) the smallest cardinality of a set of elements of G generating G together with the elements of X. The following generalizes a result originally obtained by W. Gaschütz [3] for \(X=\varnothing .\)

Lemma 11

( [1, Lemma 6]). Let X be a subset of G and N a normal subgroup of G and suppose that \(\langle g_1,\dots ,g_k, X\rangle N=G.\) If \(k\ge d_X(G),\) then there exist \(n_1,\dots ,n_k\in N,\) so that \(\langle g_1n_1,\dots ,g_kn_k,X\rangle =G.\)

It follows from Lemma 11 that the number \(\phi _{G,N}(X,k)\) of k-tuples \((g_1n_1,\dots ,g_kn_k)\) generating G with X is independent of the choice of \((g_1,\dots ,g_k).\) In particular,

$$\begin{aligned} \phi _{G,N}(X,k)=|N|^kP_{G,N}(X,k) \end{aligned}$$

where \(P_{G,N}(X,k)\) is the conditional probability that k elements of G generate G with X,  given that they generate G with XN. In particular, if

$$\begin{aligned} 1=N_0<N_1<\dots <N_t=G \end{aligned}$$

is a chief series of G,  then

$$\begin{aligned} \phi _{G,N}(X,k)=\prod _{1\le i \le t}\phi _{G/N_{i-1},N_i/N_{i-1}}(XN_{i-1},k). \end{aligned}$$

We apply the previous observations in the particular case when \(X=\{g\}\) with \(g\in V(G)\) and \(k=1.\) In this case, setting \(\delta _{G/N_{i-1},N_i/N_{i-1}}(g)=\phi _{G/N_{i-1},N_i/N_{i-1}}(gN_{i-1},1),\) we get

$$\begin{aligned} \delta _G(g)=\prod _{1\le i\le t}\delta _{G/N_{i-1},N_i/N_{i-1}}(g). \end{aligned}$$
(4.1)

Lemma 12

Let N be a minimal normal subgroup of a finite group G and let \(g\in V(G).\) If either \(N\not \le {{\,\mathrm{Frat}\,}}(G)\) or \(|N|>2,\) then \(\delta _{G,N}(g)\ge 2.\)

Proof

If \(N\le {{\,\mathrm{Frat}\,}}(G),\) then we have \(\langle g, xn \rangle =G\) whenever \(\langle g, x\rangle =G\) and \(n\in N,\) so \(\delta _{G,N}(g)=|N|.\) We may assume \(N\not \le {{\,\mathrm{Frat}\,}}(G).\) We distinguish two cases:

  1. (1)

    N is abelian. Let \(q=|{{\,\mathrm{End}\,}}_G(N)|\) and \(r=\dim _{{{\,\mathrm{End}\,}}_G(N)}N.\) By [5, Corollary 7], \(P_{G,N}(g,1)\ge \frac{q-1}{q},\) so

    $$\begin{aligned} \delta _{G,N}(g)\ge \frac{|N|(q-1)}{q}=\frac{q^r(q-1)}{q}=q^{r-1}(q-1)\ge 2, \end{aligned}$$

    except in the case \(q=2\) and \(r=1.\)

  2. (2)

    N is non-abelian. Choose x such that \(\langle g, x\rangle =G.\) Since N is non-abelian, by the main theorem in [7], there exists \(1\ne n\in N\) such that \([g,n]=1.\) Since \(N\cap Z(G)=1,\) it must be \(x^n=x[x,n]\ne x.\) On the other hand, \(G=\langle g, x \rangle = \langle g^n, x^n \rangle = \langle g, x[x,n]\rangle ,\) so the coset xN contains two different elements x and x[x, n] adjacent to g in the graph \(\Delta (G).\) This implies \(\delta _{G,N}(g)\ge 2.\)

\(\square \)

Proof of Theorem 2

Let r be the number of non-Frattini factors of order 2 in a chief series of G. Since \(C_2^r\) is an epimorphic image of G and G can be generated by 2 elements, it must be \(r\le 2.\) So the conclusion follows combining (4.1) and Lemma 12. \(\square \)

Lemma 13

Let M be a closed subgroup of a 2-generated profinite group G. If \(g\in V(G)\) and \(\langle g, x \rangle M=G,\) then there exists \(m\in M\) such that \(\langle g, xm\rangle =G.\)

Proof

Let \({\mathcal {N}}\) be the set of the open normal subgroups of G. Given \(N\in {\mathcal {N}},\) let \(\Omega _N=\{m\in M \mid \langle g,xm \rangle N=G\}.\) It follows from Lemma 11 that \(\Omega _N\ne \varnothing .\) Moreover, if \(m\in \Omega _N,\) then \(m(N\cap M)\subseteq \Omega _N\) and consequently \(\Omega _N\) is a closed subset of M. If \(N_1,\dots ,N_t\in {\mathcal {N}},\) then \(\varnothing \ne \Omega _{N_1\cap \dots \cap N_t}\subseteq \Omega _{N_1}\cap \dots \cap \Omega _{N_t}.\) Since M is compact, \(\cap _{N\in {\mathcal {N}}}\Omega _N\ne \varnothing .\) Let \(m\in \cap _{N\in {\mathcal {N}}}\Omega _N.\) Since \(\langle g,xm \rangle N=G\) for every \(N\in {\mathcal {N}},\) we have \(\langle g,xm \rangle =G.\)\(\square \)

Proof of Theorem 1

Let \(g\in V(G)\) and assume, by contradiction, that \(\delta _G(g)\) is finite. Set \(u=\delta _G(g).\) Since G is infinite, there exists an open normal subgroup N of G with the property that the length of a chief series of G is equal to \(\lceil \log _2 u\rceil +3.\) By Theorem 2, \(\delta _{G/N}(gN)\ge 2^{t-2}=2^{\lceil \log _2 u\rceil +1}\ge 2u.\) This means that there exist \(x_1,\dots ,x_{2m} \in G\) such that \(x_1N\ne \dots \ne x_{2u}N\) and \(\langle x_1, g\rangle N=\dots =\langle x_{2u},g\rangle N=G.\) By Lemma 13, there exist \(n_1,\dots ,n_{2u} \in N\) such that \(G=\langle g, x_1n_1\rangle =\dots =\langle g, x_{2u}n_{2u}\rangle .\) This implies \(u=\delta _G(g)\ge 2u,\) a contradiction. \(\square \)