Recognisability of the sporadic groups by the isomorphism types of their prime graphs

Suppose that G is a finite group, π(G) is the set of prime divisors of its order, and ω(G) is the set of orders of its elements. We define a graph on π(G) with the following adjacency relation: different vertices r and s from π(G) are adjacent if and only if rs ∈ ω(G). This graph is called the Gruenberg-Kegel graph or the prime graph of G and is denoted by GK(G). In a series of papers, we describe the coincidence conditions for the prime graphs of nonisomorphic simple groups. This issue is connected with Vasil’ev’s Question 16.26 in the Kourovka Notebook about the number of nonisomorphic simple groups with the same prime graph. Earlier the author derived necessary and sufficient conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type over fields of orders q and q1, respectively, with the same characteristic. Let G and G1 be two nonisomorphic finite simple groups of Lie type over fields of orders q and q1, respectively, with different characteristics. The author also obtained necessary conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type. In the present paper the latter result is refined in the case where G is a simple linear group of sufficiently high Lie rank over a field of order q. If G is a simple linear group of sufficiently high Lie rank, then we prove that the prime graphs of G and G1 may coincide only in one of the nineteen cases. As corollaries of the main result, we obtain constraints (under some additional conditions) on the possible number of simple groups whose prime graph is the same as the prime graph of a simple linear group.

Let Gbe a finite group and Sa sporadic simple group. We denote by π(G) the set of all primes dividing the order of G. The prime graph Γ(G) of Gis defined in the usual way connecting pand qin π(G) when there is an element of order pqin G. The main purpose of this paper is to determine finite group Gsatisfying Γ(G) = Γ(S) (See Theorem 3) and to give applications which generalize Abe (Abe, S. Two ways to characterize 26 sporadic finite simple groups. Preprint) and Chen (Chen, G. (1996). A new characterization of sporadic simple groups. Algebra Colloq.3:49–58). The results are elementary but quite useful.

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

Let G be a finite group. The prime graph Γ ( G ) of G is defined as follows: The set of vertices of Γ ( G ) is the set of prime divisors of | G | and two distinct vertices p and p ′ are connected in Γ ( G ) , whenever G contains an element of order p p ′ . A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ ( G ) = Γ ( P ) , G has a composition factor isomorphic to P. It is been proved that finite simple groups 2 D n ( q ) , where n ≠ 4 k , are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2 D 2 k ( q ) , where k ≥ 9 and q is a prime power less than 10 5 .

The prime graph, or Gruenberg–Kegel graph, of a finite group 𝐺 is the graph Γ ⁢ ( G ) \Gamma(G) whose vertices are the prime divisors of | G | \lvert G\rvert and whose edges are the pairs { p , q } \{p,q\} for which 𝐺 contains an element of order p ⁢ q pq . A finite group 𝐺 is recognisable by its prime graph if every finite group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) \Gamma(H)=\Gamma(G) is isomorphic to 𝐺. By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which 𝐺 is one of the 26 sporadic simple groups. Existing work of various authors answers the question of recognisability by prime graph for all but three of these groups, namely the Monster, M \mathrm{M} , the Baby Monster, B \mathrm{B} , and the first Conway group, Co 1 \mathrm{Co}_{1} . We prove that these three groups are recognisable by their prime graphs.

Further Reflections on Thompson's Conjecture

In this paper, we prove that there exists an infinite series of finite simple groups of Lie type with connected prime graphs which are uniquely determined by their prime graphs. More precisely, we show that every finite group G with the same prime graph as \({{}^2D_{n}(3)}\) , where n ≥ 5 is odd, is necessarily isomorphic to the group \({{}^2D_{n}(3)}\) . In fact, we give a positive answer to an open problem that arose in Zavarnitsine (Algebra Logic 45(4):220–231, 2006). As a consequence of our result, we obtain that the simple group \({{}^2D_n(3)}\) , where n is an odd number, is characterizable by its spectrum.

The prime graph of a finite group $G$ is denoted by$ga(G)$. A nonabelian simple group $G$ is called quasirecognizable by primegraph, if for every finite group $H$, where $ga(H)=ga(G)$, thereexists a nonabelian composition factor of $H$ which is isomorphic to$G$. Until now, it is proved that some finite linear simple groups arequasirecognizable by prime graph, for instance, the linear groups $L_n(2)$ and $L_n(3)$ are quasirecognizable by prime graph. In this paper, we consider thequasirecognition by prime graph of the simple group $L_n(5)$.

Let \(G\) be a finite group. Its spectrum \(\omega(G)\) is the set of all element orders of \(G\). The prime spectrum \(\pi(G)\) is the set of all prime divisors of the order of \(G\). The Gruenberg–Kegel graph (or the prime graph) \(\Gamma(G)\) is the simple graph with vertex set \(\pi(G)\) in which any two vertices \(p\) and \(q\) are adjacent if and only if \(pq\in\omega(G)\). The structural Gruenberg–Kegel theorem implies that the class of finite groups with disconnected Gruenberg–Kegel graphs widely generalizes the class of finite Frobenius groups, whose role in finite group theory is absolutely exceptional. The question of coincidence of Gruenberg–Kegel graphs of a finite Frobenius group and of an almost simple group naturally arises. The answer to the question is known in the cases when the Frobenius group is solvable and when the almost simple group coincides with its socle. In this short note we answer the question in the case when the Frobenius group is nonsolvable and the socle of the almost simple group is isomorphic to \(PSL_{2}(q)\) for some \(q\).

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pkand natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Apwith p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns: All sporadic simple groups, the alternating groups Ap (p ≥ 5 is a twin prime) and some simple groups of the Lie type. In this paper, the authors continue this investigation. In particular, the authors show that the symmetric groups Sp+3, where p + 2 is a composite number and p + 4 is a prime and 97 < p ∈ π(1000!), are 3-fold OD-characterizable. The authors also show that the alternating groups A116 and A134 are OD-characterizable. It is worth mentioning that the latter not only generalizes the results by Hoseini in 2010 but also gives a positive answer to a conjecture by Moghaddamfar in 2009.

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let L=Ln(2) or Un(2), where n≧17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that Γ(G)=Γ(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of Ln(2). Also we conclude that the simple group Un(2) is quasirecognizable by element orders.

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group G isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of G by the solvable radical of G is an almost simple group. The main goal of this paper is to prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when G is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of G, where the prime graph of G is the graph whose vertices are the prime numbers dividing the order of G and two such numbers r and s are adjacent if and only if and G has an element of order rs. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by Cameron and Maslova concerning finite groups almost recognizable by prime graph.