On Recognition by Order and Degree Pattern of Finite Simple Groups
aa r X i v : . [ m a t h . G R ] M a y On Recognition by Order and Degree Pattern ofFinite Simple Groups
B. Akbari and
A. R. Moghaddamfar
Department of Mathematics, K. N. Toosi University of Technology,P. O. Box - , Tehran, IranandE-mails : [email protected] and [email protected] June 26, 2018
Abstract
Let GK( G ) be the prime graph associated with a finite group G and D ( G ) be the degreepattern of G . A finite group G is said to be k -fold OD-characterizable if there exist exactly k non-isomorphic groups H such that | H | = | G | and D ( H ) = D ( G ). A 1-fold OD-characterizablegroup is simply called OD-characterizable. The purpose of this paper is threefold. First,it provides the reader with a few useful and efficient tools on OD-characterizability of finitegroups. Second, it lists a number of such simple groups that have been already investigated.Third, it shows that the simple groups L (3) and U (5) are OD-characterizable, too. Keywords : prime graph, degree pattern, simple group.
In this paper, we will consider only finite groups. The set of elements order of a finite group G isdenoted by ω ( G ) and called the spectrum of G (see [27]). This set is closed and partially orderedby the divisibility relation; therefore, it is determined uniquely from the subset µ ( G ) of all maximalelements of ω ( G ) with respect to divisibility. For a natural number n we denote by π ( n ) the setof prime divisors of n and put π ( G ) = π ( | G | ).There are a lot of ways to associate a graph to a finite group. One of well-known graphsassociated with a finite group G is called prime graph (or Gruenberg-Kegel graph ) of G and denotedby GK( G ), which is a simple undirected graph and construct as follows. The vertices are all primedivisors of | G | and two distinct vertices p and q are adjacent if and only if pq ∈ ω ( G ). Thenumber of connected components of GK( G ) is denoted by s ( G ), and the connected componentsare denoted by π i = π i ( G ), with 1 ≤ i ≤ s ( G ). When the group G has even order, we will assumethat 2 ∈ π ( G ). Finite simple groups with disconnected prime graph were described in [11, 24]. Acomplete list of these groups (with corrected misprints) can be found in [12, Tables 1(a)-1(c)].In [21], Suzuki studied the structure of the prime graphs associated with finite simple groups,and found the following interesting result. Proposition 1.1 (Suzuki) ([21], Theorem B) Let G be a finite simple group whose prime graph GK( G ) is disconnected and let ∆ be a connected component of GK( G ) whose vertex set does notcontain . Then ∆ is a clique. Mathematics Subject Classification : 20D05, 20D06. emark 1.1 By a clique we shall mean a complete graph and we write K n to be the clique oforder n . Let Γ = (
V, E ) be a simple graph. Note that, in Proposition 1.1, it does not require G tobe simple. Moreover, it shows that the prime graph of G has the following structure: GK( G ) = GK[ π ] ⊕ K n ⊕ · · · ⊕ K n s , where GK[ π ] denotes the induced subgraph GK( G )[ π ] , n i = | π i | and s = s ( G ) . We recall that theinduced subgraph GK( G )[ π ] has the vertex set π and it contains all edges of GK( G ) which joinvertices in π . The degree deg G ( p ) of a vertex p ∈ π ( G ) is the number of edges incident on p in GK( G ). If π ( G ) = { p , p , . . . , p k } with p < p < · · · < p k , then we putD( G ) := (cid:0) deg G ( p ) , deg G ( p ) , . . . , deg G ( p k ) (cid:1) , and call this k -tuple the degree pattern of G . We denote by h OD ( G ) the number of isomorphismclasses of finite groups H such that | H | = | G | and D( H ) = D( G ). Obviously, 1 h OD ( G ) < ∞ for any finite group G . In terms of function h OD ( · ), the groups G are classified as follows: Definition 1.1
A finite group G is called a k -fold OD-characterizable group if h OD ( G ) = k . A -fold OD-characterizable group is simply called an OD-characterizable group.
The notation for sporadic and simple groups of Lie type is borrowed from [6]. Moreover, we usethe notation A n to denote an alternating of degree n . There are scattered results in the literatureshowing that many of simple groups have the same prime graphs, for instance, it was shown in [7],[23], [35] that each of the following sets consist of simple groups(a) { S n ( q ) , O n +1 ( q ) } [23, Proposition 7.5](b) { L (11) , M } , [7, Theorem 3](c) { A , A } , { A , L (49) , U (3) } , { A , J , S (2) , O +8 (2) } , [35, Theorem](d) { A n , A n − } , n is odd, and n and n − S n ( q ) and the orthogonal group O n +1 ( q ) have the same orders, too, that is | S n ( q ) | = | O n +1 ( q ) | = 1(2 , q − q n n Y i =1 ( q i − . Therefore, in the case that S n ( q ) ≇ O n +1 ( q ) (that is q odd and n > h OD ( S n ( q )) = h OD ( O n +1 ( q )) ≥ . Notice that, we have S n (2 m ) ∼ = O n +1 (2 m ) and S ( q ) ∼ = O ( q ) (see for example [5]). There is alsoanother pair of non-isomorphic simple groups with the same order, that is L (2) ∼ = A and L (4).Actually, about simple groups with the same order, we have the following proposition (see [20]). Proposition 1.2
Every finite simple group can be determined up to isomorphism in the class offinite simple groups by its order, except exactly in the following cases:(a) L (2) ∼ = A and L (4) have the same order, i.e., , b) O n +1 ( q ) and S n ( q ) have the same order for q odd, n > . As an immediate consequence of Proposition 1.2 we have the following corollary.
Corollary 1.1 If G is a finite simple group, then there exists at most one finite simple group,non-isomorphic to G , with the same order and degree pattern as G . It was currently shown that many finite simple groups are OD-characterizable or 2-fold OD-characterizable. We have listed these groups in Table 1.
Table 1 . Some non-abelian simple groups S with h OD ( S ) = 1 or 2. S Conditions on
S h OD ( S ) Refs . A n n = p, p + 1 , p + 2 ( p a prime) 1 [16] , [19]5 n , n = 10 1 [8] , [10] , [14] , [18] , [29] n = 106 ,
112 1 [25] n = 10 2 [17] L ( q ) q = 2 , , [19] , [34] L ( q ) | π ( q + q +1 d ) | = 1 , d = (3 , q −
1) 1 [19] U ( q ) | π ( q − q +1 d ) | = 1 , d = (3 , q + 1) , q > L (9) 1 [31] U (5) 1 [32] L ( q ) q
37 1 [1] , [2] , [4] U (7) 1 [4] L n (2) n = p or p + 1 , for which 2 p − L n (2) n = 9 , ,
11 1 [9] , [15] U (2) 1 [33] R ( q ) | π ( q ± √ q + 1) | = 1 , q = 3 m +1 , m > q ) q = 2 n +1 > , [19] B m ( q ) , C m ( q ) m = 2 f > , | π (cid:0) ( q m + 1) / (cid:1) | = 1 , B ( q ) ∼ = C ( q ) | π (cid:0) ( q + 1) / (cid:1) | = 1 , q = 3 1 [3] B m ( q ) ∼ = C m ( q ) m = 2 f > , | q, | π (cid:0) q m + 1 (cid:1) | = 1 , ( m, q ) = (2 ,
2) 1 [3] B p (3) , C p (3) | π (cid:0) (3 p − / (cid:1) | = 1 , p is an odd prime 2 [3] , [19] B (5) , C (5) 2 [3] C (4) 1 [13] S A sporadic simple group 1 [19] S A simple group with | π ( S ) | = 4 , S = A S A simple group with | S | , S = A , U (2) 1 [28] S A simple C , - group 1 [16]According to the results in Table 1, h OD ( A ) = 2. In fact, another group with the same order3nd degree pattern as A is Z × J , that is | A | = | Z × J | = 2 · · · , D ( A ) = D ( Z × J ) = (2 , , , . Until recently, no example of simple group M has been found with h OD ( M ) ≥
3. So it wouldbe reasonable to ask whether it might be possible to find such a simple group. The followingquestion has been posed in [16]:
Problem 1.1
Is there a simple group which is k -fold OD-characterizable for k ≥ ? It is worthwhile to point out that according to the Corollary 1.1, if there exists a simple group,say S , with h OD ( S ) ≥
3, then among non-isomorphic groups with the same order and degreepattern as S , certainly there will be a non-simple group.In this paper especially we will concentrate on simple groups L (3) and U (5). In fact, we willshow that both groups are OD-characteizable. It is worth noting that the prime graph of L (3) isdisconnected, while the prime graph of U (5) is connected. Main Theorem.
The simple groups L (3) and U (5) are OD-characteizable. L (3) and U (5) In what follows, we will show that the projective special linear group L (3) and the projectivespecial unitary group U (5) are uniquely determined by order and degree pattern. Lemma 2.1
The following statements hold. (a) If L be the finite simple group L (3) , then (a.1) | L | = 2 · · · · · ; (a.2) µ ( L ) = { , , , , , , } ; (a.3) Out( L ) = Z × Z and s ( L ) = 2 ; (a.4) D ( L ) = (4 , , , , , . (b) If U be the finite simple group U (5) , then (b.1) | U | = 2 · · · · ; (b.2) µ ( U ) = { , , , } ; (b.3) Out( U ) = Z × Z and s ( U ) = 1 ; (b.4) D ( U ) = (3 , , , , .Proof . See [6]. (cid:3) ✉ ✉✉ ✉✉ (cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅(cid:0)(cid:0)(cid:0) ✉ Fig. 1.
The prime graph of L (3). ✉ ✉✉ ✉✉ (cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)
13 2 3 75
Fig. 2.
The prime graph of U (5). Let Γ = (
V, E ) be a graph with vertex set V and edge set E . A set I ⊆ V of vertices is saidto be an independent set of Γ if no two vertices in I are adjacent in Γ. The independence numberof Γ, denoted by α (Γ), is the maximum cardinality of an independent set among all independentsets of Γ. For convenience, we will denote α (GK( G )) as t ( G ) for a group G . Moreover, for avertex r ∈ π ( G ), let t ( r, G ) denote the maximal number of vertices in independent sets of GK( G )containing r . 4 heorem 2.1 (Theorem 1, [22]) Let G be a finite group with t ( G ) > and t (2 , G ) > , andlet K be the largest normal solvable subgroup K of G . Then the quotient group G/K is an almostsimple group, i.e., there exists a finite non-abelian simple group S such that S G/K Aut( S ) . Given a prime p , we denote by S p the set of non-abelian finite simple groups P such thatmax π ( P ) = p . Using Table 1 in [26] (see also [14, Table 4]), we have listed the non-abelian simplegroups and their orders in S , in Table 2. Table 2 . The simple groups S with π ( S ) ⊆ { , , , , , } . S | S | | Out( S ) | S | S | | Out( S ) | A · · A · · · · ·
13 2 A · · A · · · · ·
13 2 U (2) 2 · · A · · · · ·
13 2 A · · · A · · · · ·
13 2 A · · · G (2 ) 2 · · · ·
13 2 A · · · B (2 ) 2 · · · ·
13 6 A · · · ) 2 · · ·
13 3 B (2) 2 · · · L (2 ) 2 · · · ·
13 6 O +8 (2) 2 · · · L (3) 2 · ·
13 2 L (2 ) 2 · · · L (3) 2 · · ·
13 4 L (2 ) 2 · · L (3) 2 · · · ·
13 2 U (3) 2 · · L (3) 2 · · · · · U (3) 2 · · · B (3) 2 · · · ·
13 2 U (5) 2 · · · O +8 (3) 2 · · · ·
13 24 L (7) 2 · · G (3) 2 · · ·
13 2 B (7) 2 · · · C (3) 2 · · · ·
13 2 L (7 ) 2 · · · L (3 ) 2 · · · ·
13 4 J · · · L (3 ) 2 · · ·
13 6 A · · · ·
11 2 U (5) 2 · · · ·
13 4 A · · · ·
11 2 B (5) 2 · · ·
13 2 U (2) 2 · · ·
11 2 L (5 ) 2 · · ·
13 4 U (2) 2 · · · ·
11 6 L (13) 2 · · ·
13 2 L (11) 2 · · ·
11 2
Suz · · · · ·
13 2 M · · ·
11 1
F i · · · · ·
13 2 M · · ·
11 2 D (2) 2 · · ·
13 3 M · · · ·
11 2 F (2) ′ · · ·
13 2 HS · · · ·
11 2 U (2 ) 2 · · ·
13 4 M c L · · · ·
11 2
Lemma 2.2
Let G be a finite group, with | G | = p m p m · · · p m s s , where s, m , m , . . . , m s arepositive integers and p , p , . . . , p s distinct primes. Let ∆ = { p i ∈ π ( G ) | m i = 1 } , and for eachprime p i ∈ ∆ , let ∆( p i ) = { p j ∈ ∆ | j = i, p j ∤ p i − and p i ∤ p j − } . Let K be a normal slovablesubgroup of G . Then there hold. If p i ∈ ∆ divides the order of K , then for each prime p j ∈ ∆( p i ) , p j ∼ p i in GK( G ) . Inparticular, deg G ( p i ) > | ∆( p i ) | . (2) If | ∆ | = s and for all i = 1 , , . . . , s , | ∆( p i ) | = s − , then G is a cyclic group of order | G | and D( G ) = ( s − , s − , . . . , s − .Proof . see [2]. (cid:3) Theorem 2.2
The projective special linear group L (3) is OD -characterizable.Proof . Assume that G be a finite group such that | G | = | L (3) | = 2 · · · · · and D( G ) = D( L (3)) = (4 , , , , , . According to these conditions on G , the following two possibilities can occur for the prime graphof G : ✉ ✉✉ ✉✉ (cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅(cid:0)(cid:0)(cid:0) ✉
11 or ✉ ✉✉ ✉✉ (cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅(cid:0)(cid:0)(cid:0) ✉ Fig. 3.
All possibilities for the prime graph GK( G ). Let K be the largest normal solvable subgroup of G . We claim that K is a { , } -group. To provethis, we first observe that π ( K ) ∩ { , } = ∅ , since otherwise by Lemma 2.2, 5 ∼
7, which clasheswith the structure of GK( G ) shown in Fig. 3. Now, we show that π ( K ) ∩ { , } = ∅ . Supposethe contrary. Assume first that 11 ∈ π ( K ) and take P ∈ Syl ( K ). Then, by Frattini argument,we deduce that G = KN G ( P ). Therefore, the normalizer N G ( P ) contains an element of order 7,say x , and so P h x i is a subgroup of G of order 11 i · i = 1 ,
2, which is a cyclic group in bothcases. This shows that 7 ∼
11 in GK( G ), an impossibility. By a similar way we can show that13 / ∈ π ( K ). Finally K is a { , } -group.From the structure of the prime graph of G as shown in Fig. 3, we conclude that t (2 , G ) ≥ t ( G ) ≥
3. Consequently, from Theorem 2.1 we conclude that there exists a finite non-abeliansimple group S such that S ≤ G/K ≤ Aut( S ). Since S ∈ S and K is a { , } -group, we observethat the order of Aut( S ) is divisible by 5 · · · . On the other hand, using Table 2 we seethat π (Out( S )) ⊆ { , } , which forces | S | = 2 a · b · · · · , where 2 ≤ a ≤
11 and 0 ≤ b ≤ S is L (3) and since | G | = | L (3) | , we obtain | K | = 1 and G is isomorphic to L (3). (cid:3) Theorem 2.3
The projective special unitary group U (5) is OD -characterizable.Proof . Assume that G is a finite group such that | G | = | U (5) | = 2 · · · ·
13 and D( G ) = D( U (5)) = (3 , , , , . We have to show that G ∼ = U (5). First of all, from the structure of the degree pattern of G , it iseasy to see that 7 ≁
13 in GK( G ), since otherwise deg(2) ≤
2, which is impossible. In fact, thereare only two possibilities for the prime graph of G shown in Fig. 4.:6 ✉✉ ✉✉ (cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)
13 3 2 75 or ✉ ✉✉ ✉✉ (cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)
13 2 3 75
Fig. 4.
All possibilities for the prime graph GK( G ). Since S ∈ S and { , } ⊆ π ( S ), hence we obtain | S | = 2 a · b · c · ·
13, where 2 ≤ a ≤ ≤ b ≤ ≤ c ≤
6. Comparing the orders of simple groups listed in Table 2, we observethat the only possibility for S is L (3 ) or U (5). If S is isomorphic to L (3 ), then | K | is divisibleby 5 , because | Out( S ) | = 4. Let x ∈ K be an element of order 5 and P ∈ Syl ( K ). By Frattiniargument, we deduce that G = KN G ( P ). Hence the normalizer N G ( P ) contains an element oforder 13, say y , and so H := P h y i is a subgroup of G of order 5 ·
13. Now, one can easily verifythat the Sylow 5-subgroup P of H is a normal subgroup of H . As a matter of fact, H is a Frobeniusgroup with kernel P and complement h y i , because H does not contain an element of order 5 · | −
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