ON NEIGHBOURHOODS IN THE ENHANCED POWER GRAPH ASSOCIATED WITH A FINITE GROUP

Abstract

We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

1. Introduction

All groups considered in this paper are finite unless otherwise stated. To study the structure of a group, one can look at the invariants of some graphs whose vertices are the elements of the group and whose edges reveal some properties of the group itself. More precisely, if G is a group and ${\mathcal {B}}$ is a class of groups, the ${\mathcal {B}}$ -graph associated with G, denoted by $\Gamma _{{\mathcal {B}}}(G)$ , is a simple and undirected graph whose vertices are the elements of G, and there is an edge between two elements x and y of G if the subgroup generated by x and y is a ${\mathcal {B}}$ -group.

Several features of a finite group can be detected by analysing the invariants of its ${\mathcal {B}}$ -graph. We refer to [5] for a survey on this topic and to [10, 11] for related work. Recent papers deal with the investigation of the (closed) neighbourhood ${\mathcal {I}}_{\mathcal {B}}(x)$ of a vertex x in $\Gamma _{{\mathcal {B}}}(G)$ , that is, the set of all y in G such that x and y generate a ${\mathcal {B}}$ -group. When ${\mathcal {B}}$ is the class of abelian groups, then ${\mathcal {I}}_{\mathcal {B}}(x)$ coincides with the centraliser of x in G, thus ${\mathcal {I}}_{\mathcal {B}}(x)$ is a subgroup. However, in general this is not the case when ${\mathcal {B}}$ is distinct from the class of abelian groups. Nevertheless, even though ${\mathcal {I}}_{\mathcal {B}}(x)$ is not a subgroup of G in general, it can happen that the characteristics of a single neighbourhood in a ${\mathcal {B}}$ -graph could affect the structure of the whole group G. For instance, when ${\mathcal {B}}$ coincides with the class $\mathcal S$ of soluble groups, it has been shown that the combinatorial properties, as well as arithmetic ones, of ${\mathcal {I}}_{\mathcal {B}}(x)$ may force the whole group to be abelian or nilpotent (see [2, 3] for more details).

Here we start by considering the class ${\mathcal {C}}$ of all cyclic groups. Cameron in [5] calls the graph $\Gamma _{{\mathcal {C}}} (G)$ the enhanced power graph. The term enhanced power graph appears to have originated in [1]. However, this graph was first studied in [12] under the name cyclic graph. Further investigations under this name occurred in [13]. Recently, this graph has been investigated in [6–8] and there are still several open questions, as described in [15].

Our interest in $\Gamma _{{\mathcal {C}}} (G)$ chiefly concerns the cardinality of ${\mathcal {I}}_{\mathcal {C}} (x)$ , discussing the possible values that can occur for $|{\mathcal {I}}_{\mathcal {C}} (x)|$ when x belongs to a p-group G. Denote by $n_G$ the maximum of the sizes of all ${\mathcal {I}}_{\mathcal {C}}(x)$ for $x \in G \setminus \{1\}$ . Then clearly

$$ \begin{align*} \exp(G) \leq n_G \leq |G|, \end{align*} $$

where $\exp (G)$ denotes the exponent of the group G. Whenever G has a nontrivial universal vertex, that is, a nontrivial element adjacent to any element of G, $n_G=|G|$ . These groups have been characterised in the soluble case in [8]. Our first goal is to characterise p-groups G with $n_G = \exp (G)$ . Indeed we prove the following result.

Theorem 1.1. Let G be a finite p-group. Then $n_G=\exp (G)$ if and only if G is cyclic, or $\exp (G)=p$ , or G is a dihedral $2$ -group.

It is worth mentioning that a problem connected to closed neighbourhoods has been addressed in [14]. Going further, one may ask what is the second value that can occur for $n_G$ , and the answer is given by the following proposition.

Proposition 1.2. Let G be a p-group and assume $n_G> \exp (G)$ . Then we have ${n_G \geq p^{\alpha + 1} - p^{\alpha } + p^{\alpha -1}}$ .

We point out that the bound in Theorem 1.2 is sharp in some sense. Indeed, for $G=C_{p^2} \times C_p$ we have $n_G = p^{3} - p^{2} + p$ , where $C_k$ denotes the cyclic group of order k.

2. The cyclic graph

In this section we will deal with the enhanced power graph of a group, or what we like to call the cyclic graph of a group. Recall that the cyclic graph of a group G, denoted by $\Delta (G)$ , is the graph whose vertex set is $G \setminus \{1\}$ , and two distinct elements $x,y$ of G are adjacent if and only if $\langle x, y\rangle $ is cyclic. When x and y are adjacent we will write $x \sim y$ . We denote by $n_G$ the maximum of the sizes of all ${\mathcal {I}}_{\mathcal {C}} (x)$ for $x \in G \setminus \{1\}$ . We begin with the following useful lemma.

Lemma 2.1. Let p be a prime and let G be a p-group. Then there exists an element $z \in G$ of order p such that $|{\mathcal {I}}_{\mathcal {C}} (z)| = n_G$ .

Proof. Observe that there exists an element $x \in G$ such that $|{\mathcal {I}}_{{\mathcal {C}}} (x)| = n_G$ . If $o (x) = p$ , then we are done. Therefore, we assume that $o (x) = p^k$ , where k is an integer so that $k \geq 2$ . Take $z = x^{p^{k-1}}$ , and observe that x and z belong to the same connected component $\Upsilon $ in $\Delta (G)$ , and that z is the only element of order p in $\Upsilon $ . By [6, Lemma 2.2], $z \sim y$ for any element $y \in \Upsilon $ , and so $|{\mathcal {I}}_{{\mathcal {C}}} (z)| \geq |{\mathcal {I}}_{{\mathcal {C}}} (x)|=n_G$ , which implies $|{\mathcal {I}}_{{\mathcal {C}}} (z)| = n_G$ .

By Lemma 2.1 and [6, Lemma 2.2], one can easily see that $n_G = |\Upsilon | - 1$ , where $\Upsilon $ is a connected component of $\Delta (G)$ containing a vertex of degree $n_G$ .

2.1. Abelian p-groups

In this subsection, we focus on Abelian p-groups. In the next lemma, we compute $n_G$ when G is a nontrivial cyclic group.

Lemma 2.2. If G is a nontrivial cyclic group, then $n_G=|G|$ .

Proof. Let $x \in G$ such that $G= \langle x \rangle $ . Since $o(x)=|G|$ and $G \setminus \langle x \rangle = \emptyset $ , we conclude that $n_G=|G|$ .

We next compute $n_G$ when G is a p-group having exponent p.

Lemma 2.3. Let p be a prime and let G be a p-group of exponent p. Then $n_G = p$ .

Proof. If G is a cyclic group of order p, then the result follows from Lemma 2.2. Assume that G is not cyclic, and consider an element $x \in G$ such that $|{\mathcal {I}}_{{\mathcal {C}}} (x)| = n_G$ . As $o(x) = p$ , we have $n_G \geq p$ .

Now observe that if $y \in G \setminus \langle x \rangle $ , then $ \langle x, y \rangle $ is not cyclic. Indeed, arguing by contradiction, let $z \in G$ be such that $\langle x, y \rangle = \langle z \rangle $ . Since G has exponent p, there exist $i , j \in \{1, \ldots , p-1\}$ such that $x = z^i$ and $y = z^j$ . Therefore, from $(i,p) = 1$ it follows that $\langle x \rangle = \langle z^i \rangle = \langle z \rangle $ and $y \in \langle x \rangle $ , a contradiction. Hence, we conclude that $n_G = p$ .

We now show that if G is a noncyclic abelian group whose exponent is larger than p, then $n_G$ is larger than the exponent of G.

Lemma 2.4. Let p be a prime and let G be a noncyclic abelian p-group of exponent $exp(G) = p^{\alpha }$ , where $\alpha \geq 2$ . Then $n_G \geq p^{\alpha + 1} - p^{\alpha } + p^{\alpha -1}$ . As a consequence, ${n_G> \exp (G)}$ .

Proof. As G is abelian, we may assume

$$ \begin{align*}G=C_{p^{\alpha_1}} \times \cdots \times C_{p^{\alpha_r}},\end{align*} $$

where $r\geq 2$ , $1 \leq \alpha _1 \leq \cdots \leq \alpha _r=\alpha $ and $C_{p^{\alpha _i}}=\langle x_i\rangle $ is a cyclic group of order $p^{\alpha _i}$ .

If $\alpha _{r-1}=1$ , then the vertex $x_r^{p^{\alpha -1}}$ is adjacent to $p^{\alpha }-2$ nontrivial elements of $\langle x_r \rangle $ and to any element of the form $x_{r-1}^i x_r^k$ , where $i=1, \ldots , p-1$ and k is a positive integer less than $p^{\alpha }$ and coprime with p. Hence, there are precisely $p^{\alpha } - p^{\alpha - 1}$ choices for k, which implies

$$ \begin{align*}|I_{{\mathcal{C}}}(x)| \geq p^{\alpha} + (p-1)(p^{\alpha} - p^{\alpha-1}) = p^{\alpha + 1} - p^{\alpha} + p^{\alpha-1}.\end{align*} $$

If $\alpha _{r-1}>1$ , then one can consider the subgroup $\langle x_r^{p^{\alpha _{r-1} - 1}}, x_r \rangle $ , arguing as in the previous case.

We now collect these lemmas in a proposition where we note that, for an abelian p-group G, $n_G$ equals the exponent of G if and only if G is cyclic or elementary abelian.

Proposition 2.5. Let p be a prime and let G be an abelian p-group. Then ${n_G=\exp (G)}$ if and only if G is either cyclic or elementary abelian.

Proof. If G is either cyclic or elementary abelian, then the result follows from Lemmas 2.2 and 2.3. Conversely, assume that $n_G = \exp (G)$ . If G is neither cyclic nor elementary abelian, then, applying Lemma 2.4, we have $n_G> \exp (G)$ , a contradiction.

2.2. Nonabelian p-groups

We now shift our focus to nonabelian p-groups. When p is a prime, we take $\alpha $ to be an integer greater than $1$ when p is odd and an integer greater than $2$ when $p = 2$ . We denote by $M_{p^{\alpha +1}}$ the group

$$ \begin{align*} M_{p^{\alpha +1}} = \langle x,y \mid x^{p^{\alpha}}=y^p=1, \ x^y=x^{p^{\alpha-1}+1} \rangle. \end{align*} $$

Going further, we respectively denote by $D_{2^{\alpha +1}}$ , $S_{p^{\alpha +1}}$ and $Q_{2^{\alpha +1}}$ the dihedral, semidihedral and generalised quaternion groups given by the following presentations:

$$ \begin{align*} D_{2^{\alpha +1}}& = \langle x,y \mid x^{2^{\alpha}}=y^2=1, \ x^y=x^{-1} \rangle, \\ S_{p^{\alpha +1}} &= \langle x,y \mid x^{p^{\alpha}}=y^p=1, \ x^y=x^{p^{\alpha-1}-1} \rangle, \\ Q_{2^{\alpha +1}} &= \langle x,y \mid x^{2^{\alpha}-1}=y^2, \ y^4=1,\ x^y=x^{-1} \rangle. \end{align*} $$

The characterisation of nonabelian p-groups with a cyclic maximal subgroup is well known (see [9]).

Theorem 2.6. Let p be a prime and let G be a nonabelian p-group of order $p^{\alpha +1}$ with a cyclic subgroup of order $p^{\alpha }$ .

1. (i) If p is odd then G is isomorphic to $M_{p^{\alpha + 1}}$ .

2. (ii) If $p=2$ and $\alpha =2$ , then G is isomorphic to either $D_{8}$ or $Q_{8}$ .

3. (iii) If $p=2$ and $\alpha>3$ , then G is isomorphic to either $M_{2^{\alpha + 1}}$ , $D_{2^{\alpha + 1}}$ , $Q_{2^{\alpha + 1}}$ or $S_{2^{\alpha + 1}}$ .

We compute $n_G$ for nonabelian p-groups with a maximal cyclic subgroup of index p.

Proposition 2.7. Let p be a prime and let G be a p-group of order $p^{\alpha +1}$ . Assume that G has a maximal cyclic subgroup of order $p^{\alpha }$ . Then $n_G = \exp (G)$ if and only if either G is cyclic, or $\exp (G) = p$ , or $G \simeq D_{2^{\alpha +1}}$ .

Proof. If G is cyclic or $\exp (G) = p$ , then $n_G = \exp (G)$ by Lemmas 2.3 and 2.2. Moreover, if $G \simeq D_{2^{\alpha +1}}$ , then G has only one cyclic subgroup of order $2^{\alpha }$ while all the other cyclic subgroups have order $2$ , which implies $n_G = \exp (G)$ .

Now assume that $n_G = \exp (G)$ . If G is abelian then G is either cyclic or elementary abelian by Proposition 2.5. Now assume that G is neither abelian nor of exponent p. From Theorem 2.6 we have to analyse two cases. First assume that G is isomorphic to $M_{p^{\alpha + 1}}$ . Then $(yx)^p = x^{({p(p-1)}/{2})p^{\alpha -1}+p}$ , which yields a contradiction. Indeed, when p is odd, we have $(yx)^p = x^p$ and $|{\mathcal {I}}_{{\mathcal {C}}} (x^p)|> \exp (G)$ as $x^p$ is connected to every element of $\langle x \rangle $ and to every element of $\langle yx \rangle $ . If $p=2$ , then $(yx)^2 = x^{2^{\alpha - 1} +2}$ and ${\mathcal {I}}_{{\mathcal {C}}}(x^{2^{\alpha -1} + 2})$ contains more than $2^{\alpha }$ elements.

Finally, assume that $p = 2$ and G isomorphic to $S_{2^{\alpha + 1}}$ . Then $(yx)^2 = x^{2^{\alpha -1}}$ and $|{\mathcal {I}}_{{\mathcal {C}}} (yx)|> \exp (G)$ .

We are now in a position to prove Theorem 1.1.

Proof of Theorem 1.1.

By Lemmas 2.2 and 2.3 and Proposition 2.7, we only need to prove that if $n_G=\exp (G)$ then G is either cyclic, or $\exp (G)=p$ , or G is a dihedral $2$ -group. Thus, let $n_G=\exp (G)$ , and by way of contradiction assume neither that G is cyclic, nor $\exp (G) = p$ , nor G is a dihedral group of order $2^{\exp (G) + 1}$ , such that G has minimal order. Hence, there exists an element $x \in G$ such that $p < o(x) = \exp (G)$ . By Proposition 2.7, it follows that $p \cdot o(x) < |G|$ , and thus G contains a proper subgroup H such that $x \in H$ and $|H|=p\cdot o(x)$ . Then $\exp (H) = \exp (G)$ and H has a cyclic subgroup of index p. By Proposition 2.7, H is a dihedral group of order $2 \exp (G)$ since H is neither cyclic nor such that $\exp (H) = p$ . As a consequence G is a $2$ -group, and by minimality, $|G:H|=2$ . If $o(x)=4$ , then $|G|=16$ and an easy computation using GAP shows that this is a contradiction. Hence, we may assume $o(x)>4$ . Now assume that there exists an element $a \in G \setminus H$ such that $o(a)>4$ . Then $a^2 \in H$ and $o(a^2)>2$ . This implies that $a^2 \in \langle x \rangle $ and $|{\mathcal {I}}_{{\mathcal {C}}} (a^2)|> \exp (G)$ . Hence, we may assume that $o(a) \leq 4$ for all $a \in G \setminus H$ . First assume that $G \setminus H$ contains an element a of order $2$ . If a does not invert x, then $(xa)^2=xx^a$ is a nontrivial element of $\langle x \rangle $ , since $\langle x \rangle $ is normal in G. As a consequence, $|{\mathcal {I}}_{{\mathcal {C}}} ((xa)^2)|> \exp (G)$ . Now assume that $x^a=x^{-1}$ . Let $b \in H$ be such that $x^b=x^{-1}$ . Then $x^{ab}=x$ and $ab$ belongs to the centraliser in G of x. Thus, $(xab)^4=x^4 \neq 1$ , and $|{\mathcal {I}}_{{\mathcal {C}}} (x^4)|> \exp (G)$ . Therefore, we only need to address the case in which $o(a)=4$ for every $a \in G \setminus H$ . If $a^2 \in \langle x \rangle $ for some $a \in G \setminus H$ , then $|{\mathcal {I}}_{{\mathcal {C}}}(a^2)|> \exp (G)$ . This implies that $a^2 \in H \setminus \langle x \rangle $ . As a consequence $a^2$ inverts x. On the other hand, the dihedral groups have no automorphisms of order $4$ whose square inverts its element of maximal order (see, for instance, Theorem 34.8(a) of [4]). This final contradiction proves the theorem.