Within group theory, quantity is an interesting subject and also occupies an important position. Examples include the order of a group, the order of an element, the element of maximum order in a group, and the set of orders of its elements, and so on. Early results show that we can determine the type of a group using quantities alone. A celebrated result is the Feit-Thompson theorem: every finite group of odd order is solvable. Since the classification of finite simple groups was completed, the quantitative properties within simple groups have become highly interesting. In a recent paper [1], finite simple groups without elements of order 10 are obtained. And then, finite non-solvable groups without elements of order 10 and finite groups without elements of order 10 are obtained successively ([2] and [3]).

All groups in this paper are finite, and all simple groups are nonabelian. For a finite group G , π ( G ) stands for the set of prime divisors of the order of G , and ω ( G ) denotes its spectrum, namely the set of orders of elements of G . Since ω ( G ) is closed under divisibility, it is uniquely determined by the subset μ ( G ) of maximal elements of ω ( G ) under divisibility.

The spectrum ω ( G ) of a finite group G defines its Gruenberg-Kegel graph G K ( G ) . The vertex set of G K ( G ) is π ( G ) , and two primes p , q ∈ π ( G ) are adjacent whenever G contains an element of order pq. Clearly, ω ( G ) uniquely determines G K ( G ) . Hence, ω ( G ) = ω ( L ) implies G K ( G ) = G K ( L ) .

A finite group G is called quasi-recognizable by spectrum if every finite group H satisfying ω ( H ) = ω ( G ) possesses a unique nonabelian composition factor isomorphic to G . A finite group G isrecognizable by spectrum provided that any finite group H with ω ( H ) = ω ( G ) is isomorphic to G . Equivalently, G is recognizable if h ( G ) = 1 , where h ( G ) denotes the number of pairwise non-isomorphic finite groups H with ω ( H ) = ω ( G ) . By a well-known observation of W. J. Shi, a finite group admitting a nontrivial normal soluble subgroup cannot be recognizable. Therefore, the recognizability problem is of interest mainly for groups with trivial soluble radical, especially for simple and almost simple groups.

Let S be a finite simple group of Lie type of characteristic r , and letA be an abelian r -group equipped with an S-action. An element s ∈ S is said to be unisingular on A if s has a nonzero fixed point on A. The group Sis called unisingular if every element s ∈ S acts unisingularly on every finite abelian r -group A admitting an S -action.

In [4], finite simple linear groups over fields of characteristic 2 are recognized by spectrum. Applying the adjacency criterions of finite simple groups ([5] and [6]), we solve the recognizability problem for simple groups of type A p ( 5 ) , which is the goal of this paper. This work enriches the existing body of research on the recognition of finite simple groups by spectrum.

For prime p , if the Gruenberg-Kegel graph of simple group A p ( 5 ) is disconnected ([7] Table 2a-2c), then 4 | p + 1 . After treating with them separately, we draw the following conclusions:

Theorem 1.Let L = A p ( 5 ) beafinite simple groupwith disconnected Gruenberg-Kegel graph. Then L is recognizableby spectrum whenever p > 3 .

The following conclusion is an immediate consequence of the known results.

Theorem 2.Let L = A p ( 5 ) beafinite simple groupwith disconnected Gruenberg-Kegel graph. Then L is recognizableby spectrum for p > 3 , quasi-recognizableby spectrum for p = 3 , and satisfies h ( G ) = 2 for p = 2 .

This section contains several auxiliary results required in the sequel.

Gruenberg and Kegel obtained the following description of finite groups with disconnected graph G K ( G ) , which was summarized in ([8] Lemma 1).

Lemma 2.1 (Gruenberg-Kegel Theorem). If G is a finite group with disconnectedgraph G K ( G ) , then one of the following holds:

(a) s ( G ) = 2 , G is a Frobenius group;

(b) s ( G ) = 2 , G is a 2-Frobenius group;

(c) LetK denotea nilpotent normal subgroup of G . Then there exists a non-abelian simple group S satisfying S ≤ G ¯ = G / K ≤ A u t ( S ) ;moreover, K and G ¯ / S are π 1 ( G ) -groups,the graph G K ( S ) is disconnected, s ( S ) ≥ s ( G ) ,and for every i , 2 ≤ i ≤ s ( G ) ,thereexists j satisfying 2 ≤ j ≤ s ( S ) such that ω i ( G ) = ω j ( S ) .

A more constructive result, which generalizes the Gruenberg-Kegel theorem in a certain sense, was obtained in [5].

Lemma 2.2 ([5] Corollary 7.2). Let L be a finite non-abelian simple group other than A 2 ( 3 ) , A 2 3 ( 3 ) , C 2 ( 3 ) , A l t 10 and A l t n with n satisfying { r | n − 3 ≤ r ≤ n , r   i s   p r i m e } = ∅ .Let G bea finite group with ω ( G ) = ω ( L ) , andK denotethe maximal normal soluble subgroup of G. Then there exists a finite nonabelian simple group S satisfying S ≤ G ¯ = G / K ≤ A u t ( S ) . Moreover, t ( S ) ≥ t ( G ) − 1 , and one of the following statements holds:

(1) S ≅ A l t 7 or A 1 ( q ) for some odd q ,and t ( S ) = t ( 2 , S ) = 3 .

(2) Each prime p ∈ π ( G ) non-adjacent to 2 in G K ( G ) satisfies that the Sylow p-subgroup of G is isomorphic to the Sylow p-subgroup of S.In particular, t ( 2 , S ) ≥ t ( 2 , G ) .

For a finite group, we write ρ ( G ) ( ρ ( r , G ) ) for some independent set in G K ( G ) (containing r ) with a maximal number of vertices, and put t ( G ) = | ρ ( G ) | and t ( r , G ) = | ρ ( r , G ) | . Let s = s ( G ) denote the number of connected components of G K ( G ) , and let π i = π i ( G ) be its i th connected component for i = 1 , ⋯ , s . For any even order group G, we adopt the convention that 2 ∈ π 1 . For each i , let μ i = μ i ( G ) (resp. ω i = ω i ( G ) ) be the collection of all n ∈ μ ( G ) (resp. all n ∈ ω ( G ) ) such that every prime divisor of n lies in π i . The symbol [ x ] stands for the integral part of x .

Lemma 2.3 ([7], Lemma 1.1). Let G be a finite simple group whose graph G K ( G ) isdisconnected.Then μ i ( G ) is a singletonfor 2 ≤ i ≤ s ( G ) . Write n i forthe unique element in μ i ( G ) whenever i > 1 .Thegroup G , the value t ( G ) , and n i ( G ) with 2 ≤ i ≤ s ( G ) arelistedinTables1-3, where pstandsforan odd prime.

Let q be a natural number and r an odd prime satisfying ( r , q ) = 1 . We write e ( r , q ) for the minimal positive integer number n such that q n ≡ 1 ( mod r ) . If q is odd, we set e ( 2 , q ) = 1 when q ≡ 1 ( mod 4 ) and e ( 2 , q ) = 2 when q ≡ − 1 ( mod 4 ) . An odd prime r with e ( r , q ) = n is called a primitive prime divisor of q n − 1 , denoted by r n . Zsigmondy proved the existence of such a prime in [9].

Lemma 2.4([10], Lemma 1). SupposeGisa finite group and N ⊲ G . Assume G / N is a Frobenius group with kernel F and a cyclic complement C.If ( | F | , | N | = 1 ) and F ⊈ N C G ( N ) / N ,thenthere existsaprime divisor p of | N | such that p | C | ∈ ω ( G ) .

In this section, we investigate the spectrum recognizability of the simple group A p ( 5 ) whose Gruenberg-Kegel graph is disconnected, which is stated in Theorem 1.

At first we assume the following hypothesis:

Let L = A p ( 5 ) where p > 3 ,andsuppose G isa finite groupsatisfying ω ( G ) = ω ( L ) .Then G K ( G ) = G K ( L ) , t ( L ) = [ p + 2 2 ] , ρ ( 2 , L ) = { 2 , r p } , ρ ( 5 , L ) = { 5 , r p , r p + 1 } , s ( L ) = 2 and n 2 ( G ) = n 2 ( L ) = 5 p − 1 4 > 1093 .

By Lemma 2.1 and Lemma 2.2, if K denotes the maximal normal soluble subgroup of G , then there exists a finite non-abelian simple group S such that S ≤ G ¯ = G / K ≤ A u t ( S ) . Furthermore, s ( S ) ≥ s ( G ) = 2 , and n i ( S ) = n 2 ( G ) = 5 p − 1 4 for some i with 2 ≤ i ≤ s ( S ) .

By Lemma 2.3, the group G , the value t ( G ) and n i ( G ) with 2 ≤ i ≤ s ( G ) are listed in the following Table 1.

Table 1. Finite simple groups G satiffying s ( G ) ≥ 2 , t ( G ) ≥ 4 and n i ≥ 1093 .

All n i ( G ) ( 2 ≤ i ≤ s ( G ) ) are quoted from the three tables above and will not be reproduced unless required.

Suppose K ≠ 1 . For every prime r ∈ π ( K ) , we have O r ( K ) ≠ K ([6] Lemma 2.2). Let G ˜ = G / O r ( K ) and K ˜ = K / O r ( K ) denote these factor groups. Then K ˜ is a nontrivial r-group. Write ϕ ( K ˜ ) for the Frattini subgroup of K ˜ , and define the factor groups G ^ = G ˜ / ϕ ( K ˜ ) and K ^ = K ˜ / ϕ ( K ˜ ) . As G / K ≅ G ^ / K ^ , the relation ω ( G ^ ) ⊈ ω ( L ) implies ω ( G ) ⊈ ω ( L ) . It follows that we are allowed to replace G and K by G ^ and K ^ , respectively. After this reduction, we may assume that K is a nontrivial elementary abelian r -group ([4], Proposition 1).

Lemma 3.1. S is not isomorphic to any sporadic groups.

Proof. When S is a sporadic group, we have n i ( S ) ≤ 71 < 5 p − 1 4 , then S is not isomorphic to any sporadic groups by Table 1. □

Since t ( L 2 ( q ) ) = 3 < t ( L ) − 1 , we have S ≇ A 1 ( q ) . Thus the second statement of Lemma 2.2 holds, then r p ∉ π ( K ) and ρ ( 2 , G ) = { 2 , r p } ⊆ ρ ( 2 , S ) . We now examine all possibilities for the non-abelian simple group S listed in Table 1.

Lemma 3.2. S ≇ A l t n ( n ≥ 5 ) .

Proof. Let S ≅ A l t n ( n ≥ 5 ) , then at least one prime among n , n − 1 , n − 2 , n − 3 equals r p . Since n i ( S ) = n 2 ( L ) , we have n ≥ 11 and r p = 5 p − 1 4 . On the other hand, r 2 p | 5 p + 1 and 5 p + 1 ≡ 0 ( mod 4 ) , then r 2 p ≤ 5 p + 1 4 < 5 p − 1 4 − 6 ≤ r p − 6 . So r 2 p ∈ π ( S ) and r 2 p < n − 3 , then r 2 p is adjacent to 5 in G K ( S ) ([5], Proposition 1.1). However they are non-adjacent in G K ( L ) , we obtain a contradiction. □ .

Lemma 3.3. S is not isomorphic to any simple exceptional groups of Lie type.

Proof. Let S ≅ 3 D 4 ( q ) , then t ( L ) ≤ t ( S ) + 1 ≤ 4 , therefore p ≤ 5 . However in this case where n 2 ( L ) = n 2 ( S ) < 1093 , a contradiction.

Let S ≅ E 8 ( q ) , then t ( L ) ≤ t ( S ) + 1 = 13 , therefore p ≤ 17 . And since 31 ∈ π ( S ) , then p = 17 .But in this case, G K ( L ) is not disconnected since q − 1 does not divisible p + 1 , a contradiction by Table 1.

Let S ≅ E 7 ( q ) , then t ( L ) ≤ t ( S ) + 1 = 8 , therefore p ≤ 7 . If q ≥ 4 , then s ( S ) = 1 < s ( G ) , a contradiction. If q < 4 , then n i ( S ) < 1093 , we obtain a contradiction by n i ( S ) = n 2 ( L ) .

Let S ≅ G 2 ( q ) , then t ( L ) ≤ t ( S ) + 1 = 4 , therefore p = 3 or 5. By n 2 ( L ) = n i ( S ) , p = 3 and q = 4 or 3, then 21 ∉ ω ( L ) and 30 ∉ ω ( L ) . If q = 4 , then 21 ∈ ω ( S ) , a contradiction. Thus q = 3 , then 5 ∈ π ( K ) for G / K ≤ A u t ( S ) . And since G 2 ( 3 ) includes a Frobenius subgroup with kernel of order 13 and cyclic complement of order 6, then 30 ∈ ω ( G ) , a contradiction.

Let S ≅ F 4 ( q ) or E 6 ε ( q ) , then t ( L ) ≤ t ( S ) + 1 ≤ 6 , therefore p ≤ 7 . If p = 3, then 21 ∉ ω ( L ) , however 21 ∈ ω ( S ) , a contradiction. If p = 5, then 33 ∉ ω ( L ) . Since 11 = r p ∈ π ( S ) , we have 33 ∈ ω ( S ) , a contradiction. Thus p = 7, and n 2 ( L ) = 1093 . However the equation n i ( S ) = n 2 ( L ) admits no solutions ([11] Theorem 3), a contradiction.

Let S be isomorphic to a finite simple Suzuki or Ree group, then t ( L ) ≤ t ( S ) + 1 ≤ 6 , therefore p ≤ 7 . However in this case the equation n i ( S ) = n 2 ( L ) admits no solutions, a contradiction ([11], Theorem 7). □

Lemma 3.4. S is not isomorphic any group of B n ( q ) , C n ( q ) or D n ε ( q ) .

Proof. If S ≅ B n ( q ) or C n ( q ) , then S ≅ C m ( 2 ) by the condition t ( 2 , S ) ≥ 3 , with m a prime. Then n 2 ( S ) = 2 m − 1 = 5 p − 1 4 , and this equation admits no solutions whenever p > 3 ([11], Theorem 7).

If S ≅ D 2 n ( q ) , then n i ( S ) ∈ { q n + 1 , 2 n − 1 + 1 , 3 n − 1 + 1 2 , 3 n + 1 4 } byTable 1 and t ( 2 , S ) , where q is even. Obviously, when n i ( S ) ∈ { q n + 1 , 2 n − 1 + 1 , 3 n − 1 + 1 2 } , the equation n i ( S ) = n 2 ( G ) = 5 p − 1 4 has no solutions. And for n i ( S ) = 3 n + 1 4 , the equation has also no solutions ([11] Theorem 1).

If S ≅ D n ( q ) , then S ≅ D n ( 5 ) or D n ( 2 ) by Table 1 and t ( 2 , S ) . Obviously, since n i ( S ) = n 2 ( G ) , we have S ≇ D n ( 2 ) ; otherwise, 2 n − 1 = n 2 ( D n ( 2 ) ) = 5 p − 1 4 , which has no solutions whenever p > 3 ([11] Theorem 7). When S ≅ D n ( 5 ) , by n i ( S ) = n 2 ( G ) , we infer that n = p ,i.e., S ≅ D p ( 5 ) . Noticing that S < G / K ≤ A u t ( S ) implies s ( G / K ) = 1 ([10] Theorem 5), we have S = G / K . Suppose K ≠ 1 and l ∈ π ( K ) , then K can taken as an element abelian l -group. Suppose that C = C G ( K ) ≠ K . Since C is normal in G and G / K is simple, we have C / K = G / K = S . Therefore r ⋅ r p ∈ ω ( G ) = ω ( L ) . However r p is non-adjacent to any other primes in G K ( L ) , a contradiction. Thus C = K and S acts faithfully on K . If l = 5 , then S acts unisingularly on K ([12] Theorem 1.3). Denote the natural semidirect product K ⋋ S by M , then ω ( M ) ⊆ ω ( G ) . Denote by o ( x ) the order of an element x . Let x ∈ K and y ∈ S such that o ( x ) = 5 and o ( y ) = r p , then ( x , y ) ∈ M and o ( x , y ) = 5 ⋅ r p , therefore 5 r p ∈ ω ( G ) , a contradiction. If l ≠ 5 , then S includes a Frobenius subgroup with kernel of order q 2 p and cyclic complement of order r p . Therefore l ⋅ r p ∈ ω ( G ) , a contradiction. Thus K = 1 , i.e., S = G. However t ( S ) = [ 3 p + 1 4 ] and t ( G ) = p + 1 2 , we obtain a contradiction. Then S ≇ D p ( 5 ) .

Lemma 3.5. S ≅ A p ( 5 ) .

Proof. Suppose false. Let S ≅ A n ε ( q ) where q = r k and q ≠ 5 , then the group S is one of the groups in the following list by Table 1:

A 1 ( q ) ,where q > 2 and q is even; A n − 1 ( q ) ,where n is an odd prime, ( n , q ) ≠ ( 3 , 2 ) , ( 3 , 4 ) ; A n ( q ) ,where n is an odd prime, ( q − 1 ) | ( n + 1 ) ; 2 A n − 1 ( q ) ,where n is an odd prime; 2 A n ( q ) ,where n is an odd prime, ( q + 1 ) | ( n + 1 ) , ( n , q ) ≠ ( 3 , 3 ) , ( 5 , 2 ) .

In this case n − 2 ≥ p for p ≥ 11 . We have n 2 ( S ) ≥ q n − 2 2 , therefore q < 5 by n 2 ( S ) = n 2 ( G ) . When q = 2 and p ≥ 11 , we have n 2 ( S ) = 2 n − 1 or 2 n + 1 3 . If n 2 ( S ) = 2 n − 1 , then 2 n + 2 − 5 p = 3 , which is impossible. If n 2 ( S ) = 2 n + 1 3 , we have 2 n + 2 = 3 ⋅ 5 p − 7 , which is also impossible. Similarly, we infer that q ≠ 3 or 4 for p ≥ 11 . Thus we have p < 11. Let p = 5, then s ≅ A 4 ( 5 ) or A 5 ( 5 ) by n 2 ( S ) = n 2 ( G ) . However in this case, t ( 2 , S ) = 2 < t ( 2 , G ) , a contradiction. Similarly we can exclude p = 7 . Similarly S isn’t isomorphic to any other case.

When S ≅ A n ( 5 ) , by n i ( S ) = n 2 ( G ) , we infer that n = p , i.e., S ≅ A p ( 5 ) .

Thus the Lemma is proved. □

Lemma 3.6. S = G

Proof. By Lemma 2.2, we obtain S ≤ G ¯ = G / K ≤ A u t ( S ) .

Suppose K ≠ 1 , then K may be assumed to be an elementary abelian r -group with r ∈ π ( K ) . Let r = 5 , then the group A p ( 5 ) acts unisingular on K , so 5 ⋅ r p ∈ ω ( G ) , which is a contradiction. Hence r ≠ 5 , and r ⋅ r p ∈ ω ( G ) . However r p is non-adjacent to any other prime in G K ( L ) , which gives another contradiction. Therefore K = 1 , and S ≤ G ≤ A u t ( S ) .

If S < G ≤ A u t ( S ) , then there exists an order-2 graph automorphism β such that G / S ≅ 〈 β 〉 ([13] Theorem 5). Consider C S ( β ) , the centralizer of β in S. Since π ( C S ( β ) ) = π ( B p − 1 ( 5 ) ) ([5] Table 4.5.2), we deduce that 5 ⋅ r 2 p − 2 ∈ ω ( G ) , which yields a contradiction. Consequently, G = S ≅ A p ( 5 ) . □

By virtue of Lemmas 3.1-3.6 above, the proof of Theorem 1 is now complete.

Grechkoseeva and Zvezdina investigated the spectrum recognition problem for simple classical groups L 4 ( q ) and U 4 ( q ) over finite fields of odd characteristic (see [14]). They proved that every finite group isospectral to L 4 ( q ) is almost simple with socle L 4 ( q ) , so L 4 ( q ) is quasi-recognizable. In particular, L 4 ( 5 ) ≅ A 3 ( 5 ) is only known to be quasi-recognizable, and the pure spectrum-recognition remains open for this group. Mazurov showed in [10] that h ( A 2 ( 5 ) ) = 2 . Hence, the spectrum-recognition problem for all simple groups A p ( 5 ) is fully settled, and Theorem 2 is proved.