Characterizability of Finite Simple Groups by their Order Components: A Summary of Results

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1 International Journal of Algebra, Vol. 4, 2010, no. 9, Characterizability of Finite Simple Groups by their Order Components: A Summary of Results Maryam Khademi Department of Mathematics Islamic Azad University - South Tehran Branch P.O. Box 11365/4435, Tehran, Iran khademi@azad.ac.ir Abstract Let G be a finite group and let OC(G) be the set of order components of G. The number of non-isomorphic classes of finite groups H satisfying OC(G) = OC(H) is denoted by h(g). If h(g) = k then G is called a k-recognizable group by the set of its order components and if k = 1, G is called a recognizable group. The main consequence of recognizability of a group G by its order components is the validity of Thompson s conjecture for G. In this paper we consider recognizability of simple groups with exactly two components and we survey currently known results in this regard which shows that the recognition problem for these groups is completely solved. Mathematics Subject Classification: 20D06, 20D60 Keywords: prime graph, order component, Thompson s conjecture 1 Introduction Let n be a positive integer. The set of all prime divisors of n is denoted by π(n) and if G is a finite group, then we set π(g) = π( G ). The Gruenberg-Kegel graph of G, or the prime graph of G, is denoted by GK(G) and is defined as follows. The vertex set of GK(G) is π(g) and two distinct vertices p and q are adjacent

2 414 M. Khademi if and only if G contains an element of order pq. We assume that {π i 1 i s(g)} is the set of connected components of GK(G). Also, we adopt the assumption that if the order of G is even, then the notation is chosen so that 2 π 1. Clearly, the order of G can be expressed as the product of the numbers m 1, m 2,..., m s(g) where π(m i ) = π i, 1 i s(g). Based on these notations, if the order of G is even and s(g) 2, then m 2,..., m s(g) are odd numbers. The positive integers m 1, m 2,..., m s(g) are called the order components of G and the set OC(G) = {m 1, m 2,..., m s(g) } is called the set of order components of G. Definition 1. Let G be a finite group. The number of non-isomorphic finite groups with the same order components as G is denoted by h(g). If h(g) = k, then G is called a k-recognizable group by the set of its order components and if k = 1, G is simply called a recognizable group. Obviously, for any finite group G we have h(g) 1. The components of the Gruenberg-Kegel graph of GK(P ) of any non-abelian finite simple group P with GK(P ) disconnected are found in [27] and [30], from which we can find the order components of P. A motivation for characterizing finite groups by the set of their order components is the following conjecture of J. G. Thompson. Conjecture (Thompson [2]) For a finite group G let N(G) = {n N G has a conjugacy class of size n}. If Z(G) = 1 and M is a non-abelian finite simple group satisfying N(G) = N(M), then G = M. 2 Preliminary results The structure of a finite group with disconnected Gruenberg-Kegel graph is determined by [30] as follows. Proposition 1. [30] Let G be a finite group with s(g) 2. Then one of the following holds: (1) G is either a Frobenius or a 2 Frobenius group.

3 Characterizability of finite simple groups 415 (2) G has a normal series 1 H K G such that H is a nilpotent π 1 group, K is a non-abelian simple group, G is a π H K 1 group, G divides K Out( K ) and any odd order component of G is equal to one of the odd H order components of K. H Proposition 2. [30] (a) Let G be a Frobenius group of even order with kernel and components K and H, respectively. Then s(g) = 2 and the components of GK(G) are π(h) and π(k). (b) Let G be a 2 Frobenius group of even order, then s(g) = 2 and G has a normal series 1 H K G such that K H = m 2, H. G K = m 1 and G K divides K H 1 and H is a nilpotent π 1 group. Proposition 3. [30] Let G be a finite group with s(g) 2. If H G is a π 1 group, then (Π s(g) j=1,j i m j)( H 1). By Proposition 3, if H is a π 1 subgroup of G, H G, and s(g) = 2, then m 2 ( H 1), implying H 1(mod m 2 ). The following result of Zsigmondy is used to prove the main theorem of [31]. Proposition 4. Let n and a be integers greater that 1. Then there exists a prime divisor p does not divide a i 1 for all i, 1 i n, except in the following cases: (1) n = 2, a = 2 k 1, where k 2. (2) n = 6, a = 2. The prime p in Proposition 4 is called a Zsigmondy prime for a n 1. In [2], it is proved that if s(g) 3, then Thompson s conjecture holds. It is also proved that if G and M are finite groups with s(m) 2, Z(G) = 1 and N(G) = N(M), then G = M, in particular s(m) = s(g) and OC(G) = OC(M). Therefore if the simple group M with disconnected prime group is characterizable by the set of its order components, then Thompson s conjecture holds for M. Therefore, in order to complete the proof of Thompson s conjecture for the simple groups P with s(p ) 2, it is enough to prove Thompson s conjecture for s(p ) = 2.

4 416 M. Khademi In [24], it is proved that if n = 2 m 4 and q is an odd prime power, then h(c n (q)) = h(b n (q)) = 2 and if q is a power of 2, then h(c n (q)) = 1. In [11] it is shown that h(b p (3)) = h(c p (3)) = 2 where p is an odd prime number, and apart these exceptional cases the rest of simple groups P with s(p ) = 2 are recognizable by the set of their order components. In a series of articles [4, 5, 6, 29] it is proved that the sporadic groups, and finite groups P SL 2 (q), 3 D 4 (q), 2 D n (3); (9 n = 2 m + 1 not a prime) and 2 D p+1 (q); (5 < p 2 m 1) are characterized by the order components of their prime graph. The recognizability of groups L p+1 (2), 2 D p (3) (where p 5 is a prime number not of the form 2 m + 1), 2 D n (2) (where n = 2 m + 1 5), D p+1 (2), D p+1 (3) and D p (q) (where p 5 is a prime number and q = 2, 3 or 5) are proved by M. R. Darafsheh et. al. in [7, 8, 9, 10, 12]. Also, characterizability of E 6 (q), 2 E 6 (q), 2 D n (q) (where n = 2 m ), P SL(p, q), P SU(p, q), P SL(p + 1, q), P SU(p + 1, q), P SL(3, q) where q is an odd prime power, P SL(3, q) for q = 2 n and P SU(3, q) for q > 5 by their order components are proved in a series of articles by B. Khosravi et. al. [18, 19, 20, 22, 23, 25, 26, 14, 15, 16]. In addition, r recognizability of B n (q) and C n (q) (where n = 2 m 4) are proved in [24]. The following open problem contains all remaining cases to prove that all simple non-abelian groups, as P, with s(p ) = 2 are characterizable by order components. Open Problem. Are the groups F 4 (q) (q odd), G 2 (q) (2 < q ±1(mod 3)) and C p (2) characterizable by their order components? Acknowledgement. The author would like to thank professor M. R. Darafsheh for introducing the subject to her, his encouragement and his invaluable comments. Also, she would like to thank the research council of the Islamic Azad University (South Tehran Branch) for supporting this research.

5 Characterizability of finite simple groups 417 Table. The simple non-abelian groups P with s(p ) = 2 and results about their characterizability by order components (p is a prime number) A n, 6 < n = p, p + 1, p + 2 one of n, n 2 is not a prime [1] A p 1 (q)(p, q) (3, 2), (3, 4) [22] A p (q)(q 1) (p + 1)) [25],[7] 2 A p 1 (q) [23] 2 A p (q)(q + 1) (p + 1), (p, q) (3, 3), (5, 2) [26] B n (q), n = 2 m q odd, B p (3), [11] C n (q), n = 2 m, C p (3) [24] C p (2) unknown D p+1 (q), q = 2, 3 [10] 2 D n (q), n = 2 m [20] 2 D n (2), n = 2 m [9] 2 D p (3), p 2 m [8] 2 D n (3), 9 n = 2 m + 1 prime [6] G 2 (q), 2 < q ±1(mod 3) unknown 3 D 4 (q) [5] F 4 (q), q odd unknown E 6 (q) [18] 2 E 6 (q) [19] 2 D p+1 (2), 5 p 2 m 1 [29] D p (q), q = 2, 3, 5 [12] M 12, J 2, Ru, He, Mcl, Co 1, Co 3, F i 22, HN [3]

6 418 M. Khademi References [1] S.H. Alavi and A. Daneshkhah, A new characterization of alternating and symmetric groups, J. Appl. Math. Comput, Vol. 17 (2005), No. 1-2, [2] Guiyun Chen, On Thompson s Conjecture, J. Algebra, 185 (1996), [3] Guiyun Chen, A new characterization of sporadic groups, Algebra colloq., 3(1) (1996), [4] Guiyun Chen, A new characterization of P SL 2 (q), Southeast Asian Bull. Math., 22 (1998), [5] Guiyun Chen, Characterization of 3 D 4 (q), Southeast Asian Bulletin of Math., 25 (2001), [6] Guiyun Chen and H. Shi, 2 D n (3)(9 n = 2 m + 1 not a prime) can be characterized by its order components, J. Appl. Math. Comput., 19(1-2) (2005), [7] M.R. Darafsheh and A. Mahmiani, A quantitative characterization of the linear groups L p+1 (2), Kumamoto J. Math., 20 (2007), [8] M.R. Darafsheh, Characterzablity of the group 2 D p (3) by its order components, where p 5 is a prime number not of the form 2 m +1, Acta Math. Sin., (Engl. Ser) 24 (7) (2008), [9] M.R. Darafsheh and A. Mahmiani, A characterization of the group 2 D n (2) where n = 2 m + 1 5, J. Appl. Math. Comput., (to appear). [10] M.R. Darafsheh, Characterization of the groups D p+1 (2) and D p+1 (3) using order components, J. Korean Math. Soc., (to appear). [11] M.R. Darafsheh, On non-isomorphic groups with the same set of order components, J. Korean Math. soc., 45 (1) (2008), [12] M.R. Darafsheh and M.Khademi, Characterization of the groups by order components, where is a prime and q = 2, 3 or 5, (manuscript). [13] M.R. Darafsheh and M.Khademi, Recognition of the groups by order components of prime graph, 40th Annual Iranian Mathemtics Conference, Sharif University of Technology (1388).

7 Characterizability of finite simple groups 419 [14] A. Iranmanesh, S.H. Alavi and B. Khosravi, A characterization of P SL(3, q), where q is an odd prime power, J. Pure Appl. Algebra 170 (2-3)(2002), [15] A. Iranmanesh, S.H. Alavi and B. Khosravi, A characterization of P SL(3, q) for q = 2 n, Acta Math. Sin (Engl. ser.), 18(3) (2002), [16] A. Iranmanesh, B. Khosravi and S.H. Alavi, A characterization of P SU(3, q) for q > 5, Southeast Asian Bull. Math. 26(2)(2002), [17] C. Jansen, K. Lux, R. parker, An atlas of Brauer characters, Oxford University Prtess, New York, [18] Behrooz Khosravi and Bahnam Khosravi, A characterization of E 6 (q), Algebras, Groups and Geometries, 19 (2002), [19] Behrooz Khosravi and Bahnam Khosravi, A characterization of 2 E 6 (q), Kumamoto J. Math., 16 (2003), [20] A. Khosravi and B. Khosravi, A characterization of 2 D n (q) where n = 2 m, Int. J. Math., Game theory and algebra, 13 (2003), [21] A. Khosravi and B. Khosravi, A new characterization of some alternating and symmetric groups, Int. J. Math. Math. Sci., 45 (2003), [22] A. Khosravi and B. Khosravi, A new characterization of P SL(p, q), Comm. Alg., 32 (2004), [23] Bahman Khosravi, Behnam Khosravi and Behrooz Khosravi, A new characterization of P SU(p, q), Acta Math. Hungar., 107 (3) (2005), [24] A. Khosravi and B. Khosravi, r recognizability of B n (q) and C n (q) where n = 2 m 4, Journal of pure and applied alg., 199 (2005) [25] Behrooz Khosravi, Bahman Khosravi and Behnam Khosravi, Characterizability of P SL(p + 1, q) by its order components, Houston Journal of Mathematics, 32 (3) (2006), [26] A. Khosravi and B. Khosravi, Charactenzability of P SU(p + 1, q) by its order components, Rocky mountain J. Math., 36 (5) (2006),

8 420 M. Khademi [27] A.S. Kondratev, On prime graph components of finite simple groups, Math. USSR Sbornik, 190 (6) (1989), [28] A.S. Kondratev and V.D. Mazurov, Recognition of alternating groups of prime degree from their element orders, Siberian Math. J., 41 (2000), [29] H. Shi and G.y. Chen, 2 D p+1 (5 < p 2 m 1) can be characterized by its order componants, Kumamoto J. Math., 18 (2005), 1 8. [30] J.S. Williams, Prime graph components of finite groups, J. Algebra, 69 (1981), [31] K. Zsigmondy, Zur theorie der potenzreset, Monatsh Math. Phy., 3 (1892), Received: October, 2009