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

Theorem 1.1

A sporadic finite simple group 𝐺 is recognisable by the isomorphism type of its prime graph if and only if 𝐺 is one of the following eight groups:

Moreover, for every sporadic finite simple group 𝐺 not appearing in the above list, there are infinitely many pairwise non-isomorphic groups 𝐻 with Γ ⁢ ( H ) ≅ Γ ⁢ ( G ) .

Remark 1.2

It is asserted in [17] that the sporadic group Ru is recognisable by its labelled prime graph. This is incorrect. There are infinitely many groups with the same prime graph as Ru . A proof is given in Section 3.7. The revised solution to the recognisability-by-labelled-prime-graph problem for the sporadic groups is summarised in Table 3.

Theorem 2.1

Theorem 2.1 (Gruenberg–Kegel)

If 𝐻 is a finite group with disconnected prime graph, then one of the following statements holds.

1. 𝐻 is a Frobenius group and Γ ⁢ ( H ) consists of two connected components Γ ⁢ ( K ) and Γ ⁢ ( C ) , where 𝐾 and 𝐶 are the Frobenius kernel and complement of 𝐻, respectively; Γ ⁢ ( K ) is a complete graph; and Γ ⁢ ( C ) is either a complete graph, or 2 , 3 , 5 ∈ π ⁢ ( C ) and Γ ⁢ ( C ) is obtained from the complete graph on π ⁢ ( C ) by deleting the edge { 3 , 5 } .

2. 𝐻 is a 2-Frobenius group, i.e. 𝐻 has shape ( A : B ) : C and both A : B and B : C are Frobenius groups. In this case, Γ ⁢ ( H ) consists of two connected components Γ ⁢ ( A ⁢ C ) and Γ ⁢ ( B ) , both of which are complete graphs.

3. There exists a non-abelian simple group 𝑆 such that

   S ⊴ H ̄ : = H / F ( H ) ≤ Aut ( S ) .

   Moreover, Γ ⁢ ( S ) is disconnected with s ⁢ ( S ) ≥ s ⁢ ( H ) , and both π ⁢ ( F ⁢ ( H ) ) and π ⁢ ( H ̄ / S ) are (possibly empty) subsets of π 1 ⁢ ( H ) .

Proof

The version of the Gruenberg–Kegel Theorem proved by Williams [31, Theorem A] asserts that either (i)′ 𝐻 is a Frobenius group, (ii)′ 𝐻 is a 2-Frobenius group, or (iii)′ there exist a nilpotent normal subgroup 𝑁 of 𝐻 and a non-abelian simple group 𝑆 such that S ⁢ ⊴ ⁢ H / N ≤ Aut ⁡ ( S ) and π ⁢ ( N ) and π ⁢ ( ( H / N ) / S ) are both subsets of π 1 ⁢ ( H ) (which does indeed contain 2 because 𝐻 must have even order, given that it has a non-abelian composition factor [13]). In case (iii)′, it follows that H / F ⁢ ( H ) is also almost simple with socle 𝑆 because N ≤ F ⁢ ( H ) , so we can replace 𝑁 by F ⁢ ( H ) to obtain case (iii) as stated. It remains to justify the claims made in (i) and (ii) about the connected components of Γ ⁢ ( H ) in (i)′ and (ii)′. These claims are proved in [34, Lemma 3]. ∎

Theorem 2.2

If 𝐻 is a finite group with t ⁢ ( H ) ≥ 3 and t ⁢ ( 2 , H ) ≥ 2 , then all of the following statements hold.

1. There exists a non-abelian simple group 𝑇 such that

   T ⊴ H ̂ : = H / R ( H ) ≤ Aut ( T ) .

2. For every coclique ρ ⊆ π ⁢ ( H ) in Γ ⁢ ( H ) of size at least 3, at most one prime in 𝜌 divides | R ⁢ ( H ) | ⋅ | H ̂ / T | = | H | / | T | . In particular, t ⁢ ( T ) ≥ t ⁢ ( H ) − 1 .

3. One of the following statements holds.

   1. Every prime r ∈ π ⁢ ( H ) that is not adjacent to 2 in Γ ⁢ ( H ) does not divide | R ⁢ ( H ) | ⋅ | H ̂ / T | = | H | / | T | . In particular, t ⁢ ( 2 , T ) ≥ t ⁢ ( 2 , H ) .

   2. There exists a prime r ∈ π ⁢ ( R ⁢ ( H ) ) that is not adjacent to 2 in Γ ⁢ ( H ) . In this case, t ⁢ ( H ) = 3 , t ⁢ ( 2 , H ) = 2 , and T ≅ Alt 7 or PSL 2 ⁡ ( q ) for some odd 𝑞.

Remark 2.3

Suppose that 𝐻 is a group with disconnected prime graph and that we have managed to deduce that 𝐻 is in case (iii) of Theorem 2.1, so that the quotient of 𝐻 by its Fitting subgroup F ⁢ ( H ) is almost simple with socle 𝑆 for some non-abelian simple group 𝑆. If we instead take the quotient of 𝐻 by the potentially larger soluble normal subgroup R ⁢ ( H ) , then we again obtain an almost simple group with the same socle 𝑆. If 𝐻 also satisfies the hypotheses of Theorem 2.2, then it follows that the non-abelian simple group 𝑇 arising in Theorem 2.2 must be isomorphic to 𝑆. (Note that the hypotheses of Theorem 2.2 do not guarantee that 𝐻 is in case (iii) of Theorem 2.1, because 𝐻 could be a Frobenius group with Frobenius complement 𝐶 such that Γ ⁢ ( C ) is not a complete graph.)

Theorem 2.4

If 𝑆 is a finite simple group with s ⁢ ( S ) ≥ 3 , then either

• s ⁢ ( S ) = 3 and 𝑆 appears in Table 1,

• s ⁢ ( S ) = 4 and 𝑆 appears in Table 2,

• s ⁢ ( S ) = 5 and S = E 8 ⁢ ( q ) with q ≡ 0 , ± 1 ⁢ ( mod ⁢ 5 ) , or

• s ⁢ ( S ) = 6 and S = J 4 .

Definition 2.5

Let 𝑞 be a positive integer. If 𝑟 is an odd prime coprime to 𝑞, then we define e ⁢ ( r , q ) to be the multiplicative order of 𝑞 modulo 𝑟, i.e. the least positive integer 𝑛 such that 𝑟 divides q n − 1 . If 𝑞 is odd, then e ( 2 , q ) : = 1 if q ≡ 1 ⁢ ( mod ⁢ 4 ) , and e ( 2 , q ) : = 2 if q ≡ − 1 ⁢ ( mod ⁢ 4 ) . A prime 𝑟 is said to be a primitive prime divisor of q n − 1 if e ⁢ ( r , q ) = n .

Lemma 2.6

Lemma 2.6 (Bang–Zsigmondy [2, 35])

If q ≥ 2 and n ≥ 1 are integers, then there exists a primitive prime divisor of q n − 1 unless ( q , n ) = ( 2 , 6 ) , ( 2 , 1 ) , or ( 3 , 1 ) .

Lemma 2.7

If S = J 4 , then Γ ⁢ ( S ) is the following graph.

In particular, | π ⁢ ( S ) | = 10 .

Proof

This is readily verified using the character table of J 4 ; see e.g. [10, pp. 188–198]. ∎

Lemma 2.8

If S = PSL 2 ⁡ ( q ) with q ≥ 4 , then Γ ⁢ ( S ) consists of three cliques. In particular, the maximal size of a coclique in Γ ⁢ ( S ) is 3, i.e. t ⁢ ( S ) = 3 .

Proof

Theorem 2.4 tells us that Γ ⁢ ( S ) has three connected components (see Table 1). It remains to confirm that each one is a clique. The maximal subgroups of 𝑆 are well known; a convenient reference is [15]. If 𝑞 is even, then the connected components of Γ ⁢ ( S ) are { 2 } and the sets of prime divisors of q ± 1 . Because 𝑆 contains maximal subgroups D 2 ⁢ ( q ± 1 ) , it contains cyclic subgroups of order q ± 1 , so the latter two components are cliques. Suppose that q ≠ 3 is a power of an odd prime 𝑝. If q ≡ 1 ⁢ ( mod ⁢ 4 ) , then the connected components of Γ ⁢ ( S ) are { p } and the sets of prime divisors of q + 1 2 and q − 1 , the latter of which contains 2. If q = 9 , then each component is an isolated vertex. If q ≠ 9 , then 𝑆 contains maximal subgroups D q ± 1 and hence cyclic subgroups of order q ± 1 2 . In particular, the second connected component is a clique. The third connected component is also a clique: the odd primes dividing q − 1 form a clique by the existence of the cyclic subgroup of order q − 1 2 , and they are all adjacent to 2 because the centraliser of an involution in 𝐺 is isomorphic to D q − 1 . If q ≡ − 1 ⁢ ( mod ⁢ 4 ) , then for q ∉ { 7 , 11 } , the same argument applies if we swap the + and − signs. If q ∈ { 7 , 11 } , then each component is an isolated vertex. ∎

Lemma 2.9

If S = P ⁢ Ω 2 ⁢ p − ⁡ ( 3 ) with p = 2 m + 1 a prime, m ≥ 1 , then

Proof

By Theorem 2.4 (Table 1), Γ ⁢ ( S ) has three connected components. The component π 1 ⁢ ( S ) consists of all of the prime divisors of

By Lemma 2.6, each factor in the product ∏ i = 1 p − 2 ( 3 2 ⁢ i − 1 ) has a primitive prime divisor. Together with the prime 3, this gives us at least p − 1 primes in π 1 ⁢ ( S ) . (Note that 2 is a primitive prime divisor of 3 2 − 1 by Definition 2.5.) Considering also the other two connected components of Γ ⁢ ( S ) , we conclude that there are at least ( p − 1 ) + 2 = p + 1 primes in π ⁢ ( S ) . ∎

Lemma 2.10

If S = E 8 ⁢ ( q ) , then | π ⁢ ( S ) | ≥ 16 , with strict inequality for q > 2 .

Proof

See [8, Lemma 2.9]. ∎

Lemma 2.11

If S = G 2 ⁢ ( 3 k ) , k ≥ 1 , then π ⁢ ( S ) consists of three cliques. In particular, the maximal size of a coclique in Γ ⁢ ( S ) is 3, i.e. t ⁢ ( S ) = 3 .

Proof

This follows from [22, Lemma 3.4]; see also [29, 30]. ∎

Lemma 2.12

If S = F 4 ⁢ ( q ) with 𝑞 even, then the following statements hold.

1. | π ⁢ ( S ) | = 6 and 8 for q = 2 and 4, respectively, and | π ⁢ ( S ) | ≥ 11 otherwise.

2. If q ≥ 4 , then Γ ⁢ ( S ) has at least three vertices of degree at least 5. In particular, Γ ⁢ ( S ) has at least 12 edges.

Proof

Assertion (ii) follows from [22, Lemma 3.1]; see also [29, 30]. Assertion (i) can be verified directly from the order formula

for q = 2 k with k ≤ 6 . Now suppose that k ≥ 7 . By Table 1, the connected component π 1 ⁢ ( S ) of Γ ⁢ ( S ) consists of all of the prime divisors of q ⁢ ( q 4 − 1 ) ⁢ ( q 6 − 1 ) . Note that ϕ i : = 2 i − 1 divides q ⁢ ( q 4 − 1 ) ⁢ ( q 6 − 1 ) for i = 2 , 3 , 4 , k , 2 ⁢ k , 4 ⁢ k , 3 ⁢ k , 6 ⁢ k . Because k ≥ 7 , the eight ϕ i are distinct, and Lemma 2.6 implies that each of them has a primitive prime divisor. Together with the prime 2, this gives us at least nine primes in π 1 ⁢ ( S ) . Considering also the other two connected components of Γ ⁢ ( S ) yields | π ⁢ ( S ) | ≥ 11 . ∎

Lemma 2.13

If S = F 4 2 ⁢ ( q ) with q = 2 2 ⁢ m + 1 , m ≥ 1 , then | π ⁢ ( S ) | ≥ 8 and every coclique in Γ ⁢ ( S ) has size at most 5, i.e. t ⁢ ( S ) ≤ 5 . Moreover, if m ≥ 2 , then Γ ⁢ ( S ) contains the following graph as a subgraph for some p 1 , … , p 6 ∈ π ⁢ ( S ) ∖ { 2 , 3 } .

Proof

This follows from [22, Lemma 3.3], but we clarify some details; see also [29, 30]. The graph pictured above is the so-called “compact form” of Γ ⁢ ( S ) . This means that each vertex other than 2 and 3 in the above graph represents a certain set of prime divisors of | S | that happen to form a clique in Γ ⁢ ( S ) , and an edge indicates that all edges from one clique to another are present in Γ ⁢ ( S ) . In particular, every coclique in Γ ⁢ ( S ) has size at most 5. Specifically, p 1 , … , p 6 represent the following sets of prime divisors of | S | :

• p 1 and p 2 represent the sets of prime divisors of q 2 ± 2 ⁢ q 3 + q ± 2 ⁢ q + 1 ;

• p 3 represents the set of prime divisors of q 2 + 1 ;

• p 4 represents the set of prime divisors of q − 1 ;

• p 5 represents the set of prime divisors of q + 1 other than 3; and

• p 6 represents the set of prime divisors of q 2 − q + 1 other than 3.

Note that these sets are disjoint, and that the sets represented by p 1 , p 2 , p 3 , and p 4 are non-empty for q = 2 2 ⁢ m + 1 , m ≥ 1 . The set represented by p 6 is also non-empty. Indeed, q 6 − 1 = ( q 3 − 1 ) ⁢ ( q − 1 ) ⁢ ( q 2 − q + 1 ) , so q 2 − q + 1 is divisible by a primitive prime divisor of q 6 − 1 = 2 6 ⁢ ( m + 1 ) − 1 , which exists by Lemma 2.6 and is not equal to 3 because 3 is a primitive prime divisor of 2 2 − 1 . The set represented by p 5 is non-empty if and only if m ≥ 2 . Indeed, this set is empty if and only if 2 2 ⁢ m + 1 + 1 = 3 k for some 𝑘, and the main result of [14] shows that this occurs if and only if k = 2 and m = 1 . Therefore, if m ≥ 2 , then every “vertex” in the compact form of Γ ⁢ ( S ) is non-empty, so Γ ⁢ ( S ) contains a subgraph of the indicated form; in particular, | π ⁢ ( S ) | ≥ 8 . If m = 1 , then Γ ⁢ ( S ) is the graph

which has 8 vertices and no coclique of size larger than 4. ∎

Lemma 2.14

Let 𝐺 be a finite group, 𝐹 a field of characteristic p > 0 , and 𝑉 an irreducible F ⁢ G -module with corresponding Brauer character 𝜒. If g ∈ G has prime order not equal to 𝑝, then the fixed-point space of 𝑔 on 𝑉 has dimension 1 | g | ⁢ ∑ x ∈ ⟨ g ⟩ χ ⁢ ( x ) .

Proof

Consider the restriction of 𝑉 to H = ⟨ g ⟩ , with its corresponding Brauer character χ H . Because | H | is coprime to 𝑝, the dimension of the fixed-point space of 𝑔 is precisely the multiplicity of the trivial character 1 H as a constituent of χ H . This is equal to the inner product of χ H and 1 H , namely

Lemma 2.15

If S = B 2 2 ⁢ ( 2 2 ⁢ m + 1 ) , m ≥ 1 , then Γ ⁢ ( S ) consists of four cliques, one of which is the isolated vertex 2. In particular, the maximal size of a coclique in Γ ⁢ ( S ) is 4, i.e. t ⁢ ( S ) = 4 . Moreover, Γ ⁢ ( S ) consists of four isolated vertices if and only if m ∈ { 1 , 2 } .

Proof

Note that Theorem 2.4 already asserts that Γ ⁢ ( S ) has four connected components; see Table 2. The order of 𝑆 is q 2 ⁢ ( q − 1 ) ⁢ ( q + 2 ⁢ q + 1 ) ⁢ ( q − 2 ⁢ q + 1 ) , where q = 2 2 ⁢ m + 1 . These four factors of | S | , call them k i for i ∈ { 1 , 2 , 3 , 4 } , are pairwise coprime, so prime divisors of distinct k i are non-adjacent in Γ ⁢ ( S ) . On the other hand, by [4, Table 8.16], 𝑆 contains a cyclic subgroup (indeed, a maximal torus) of order k i for each i ∈ { 2 , 3 , 4 } , so prime divisors of a given k i are adjacent. The final assertion follows from [6, Theorem 1]. ∎

Lemma 2.16

If S = B 2 2 ⁢ ( 2 2 ⁢ m + 1 ) , m ≥ 1 , and 𝑉 is an irreducible module for 𝑆 over a finite field F r with 𝑟 coprime to | S | , then for every prime 𝑝 dividing | S | , there exists an element of order 𝑝 in 𝑆 that fixes a non-zero vector in 𝑉.

Proof

We apply Lemma 2.14 with G = S and F = F r . Because 𝑟 is coprime to | S | , the Brauer character table of 𝑆 in characteristic 𝑟 is identical to the ordinary character table of 𝑆. The ordinary character table of 𝑆 was computed by Suzuki [26], and is conveniently reproduced by Martineau [21]. A routine calculation establishes the result. ∎

Lemma 2.17

Let q = 3 2 ⁢ m + 1 , m ≥ 1 . If Q ± : = q ± 3 ⁢ q + 1 is a prime power, then it is a prime. Moreover, if both of Q ± are prime powers (hence primes), then m ≡ 1 ⁢ ( mod ⁢ 3 ) .

Proof

The proof is adapted from [24]. Suppose that one of Q ± = p k for some prime 𝑝 and some k ≥ 1 . Note that p ≠ 3 , and that 3 m + 1 ⁢ ( 3 m ± 1 ) = p k − 1 , so in particular

If 𝑘 is even, then 3 m + 1 divides one of p k / 2 ± 1 . This means that either p k − 1 > 3 2 ⁢ m + 2 , contradicting the above inequality, or 3 m + 1 = p k / 2 + 1 , implying that 3 m ± 1 = 3 m + 1 − 2 , which is impossible. Therefore, 𝑘 is odd. If p ≡ 2 ⁢ ( mod ⁢ 3 ) , then p k − 1 ≡ 1 ⁢ ( mod ⁢ 3 ) , a contradiction. Hence p − 1 = 2 ⁢ s ⋅ 3 t for some t ≥ 1 and some 𝑠 coprime to 3. The well-known “lifting the exponent lemma” [7, Lemma A.4] therefore implies that k = 3 a ⁢ b , where a = m + 1 − t and 𝑏 is coprime to 3. Because 3 m + 1 > p k / 2 (as above), we have

Because t ≥ 1 , this implies that a / t > 3 a ⁢ b / 2 − 1 , and hence that a > 3 a / 2 − 1 . Therefore, a ≤ 1 . If a = 1 , then 1 + t > 3 b ⁢ t / 2 , so b = t = 1 because b ≠ 0 . The inequality

from (2.1) then implies that 4 3 > log 3 ⁡ ( 6 ⁢ s ) ≥ log 3 ⁡ ( 6 ) ≈ 1.63 , a contradiction. Therefore, a = 0 , so (2.1) implies that t > b ⁢ t / 2 . Hence b = 1 and k = 3 a ⁢ b = 1 , so Q ± = p k is a prime.

It remains to prove the final assertion. Recall that q = 3 2 ⁢ m + 1 . One can check that Q + ⁢ Q − = q 2 − q + 1 is divisible by 7 if and only if m ≢ 1 ⁢ ( mod ⁢ 3 ) . Therefore, if Q ± are both prime powers (hence primes), then either m ≡ 1 ⁢ ( mod ⁢ 3 ) , or m ≢ 1 ⁢ ( mod ⁢ 3 ) and Q − = 7 . The latter case implies that m = 0 , contradicting our assumption that m ≥ 1 , so the former case holds as asserted. ∎

Lemma 2.18

If S = G 2 2 ⁢ ( 3 2 ⁢ m + 1 ) , m ≥ 1 , then every coclique in Γ ⁢ ( S ) has size at most 5, i.e. t ⁢ ( S ) ≤ 5 . Moreover, if m ≥ 2 and Γ ⁢ ( S ) contains two isolated vertices, then | π ⁢ ( S ) | ≥ 9 .

Proof

The fact that every coclique in Γ ⁢ ( S ) has size at most 5 can be seen from the compact form of Γ ⁢ ( S ) (as defined in the proof of Lemma 2.13), which, per [33, Figure 1], is isomorphic to the graph shown below; see also [30, Table 2].

It remains to prove the second assertion. Suppose that m ≥ 2 and that Γ ⁢ ( S ) contains two isolated vertices. The latter assumption implies that the connected components of Γ ⁢ ( S ) consisting of the prime divisors of q ± 3 ⁢ q + 1 must be isolated vertices. (This is immediate from Table 1, given that the remaining connected component π 1 ⁢ ( S ) clearly contains at least the vertices 2 and 3.) Lemma 2.17 therefore implies that m ≡ 1 ⁢ ( mod ⁢ 3 ) . Hence m = 3 ⁢ k + 1 , where k ≥ 1 because we are assuming that m ≥ 2 . Per Table 1, the connected component π 1 ⁢ ( S ) consists of 2, 3, and the odd prime divisors of q 2 − 1 . We need to show that q 2 − 1 is divisible by at least five distinct odd primes. Given that m = 3 ⁢ k + 1 with k ≥ 1 , we have 2 ⁢ m + 1 = 3 ⁢ ℓ , where ℓ = 2 ⁢ k + 1 ≥ 3 . Therefore, q 2 − 1 = 3 6 ⁢ ℓ − 1 , which is divisible by ϕ i = 3 i − 1 for all i ∈ { 3 , ℓ , 2 ⁢ ℓ , 3 ⁢ ℓ , 6 ⁢ ℓ } . Each of these ϕ i has a primitive prime divisor by Lemma 2.6, and none of these primitive prime divisors is equal to 2 because 2 is a primitive prime divisor of 3 2 − 1 . Therefore, q 2 − 1 is certainly divisible by at least five distinct primes, except possibly when ℓ = 3 . For ℓ = 3 , namely q = 3 9 , a direct calculation shows that q 2 − 1 has exactly five distinct odd prime divisors. ∎

Remark 2.19

Throughout the proof of Theorem 1.1, we refer to the ordinary and/or modular character tables of various (usually simple) groups 𝐺. These are used together with Lemma 2.14 to determine whether certain elements of 𝐺 act fixed-point freely on the non-zero vectors of certain 𝐺-modules. In all cases that arise, this question can be answered by inspection of the character table, i.e. no computer-assisted calculation is needed to determine whether the sum appearing in Lemma 2.14 is zero. The character tables themselves are obtained from the GAP Character Table Library [5, 36] using the functions CharacterTable and/or BrauerTable (but of course, many of the ordinary character tables also appear in the Atlas [10]). For example, in Section 3.7, the 2-modular character table of Ru is obtained via the command BrauerTable("Ru",2), and in Section 4.3, the ordinary character table of E 6 2 ⁢ ( 2 ) is obtained via CharacterTable("2E6(2)"). There is one case in which we instead need to use Magma [3]. Namely, in Section 3.3, the 2-modular character table of PSL 2 ⁡ ( 61 ) is obtained via the command BrauerCharacterTable(PSL(2,61),2). We also use Magma to construct certain modules for the Suzuki groups; see Sections 3.4– 3.6.