1 / 44

Arithmetical Properties of Finite Groups

Arithmetical Properties of Finite Groups. Wujie Shi ( 施武杰）. Suzhou University, China. ISCHIA GROUP THEORY 2010 April 2010, ITALY. In memory of Maria Silvia Lucido (1963-2008), 2005 St Andrews. 0. Introduction. Let G be a finite group and Ch i ( G ) be one of the following sets:

prue
Download Presentation

Arithmetical Properties of Finite Groups

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Arithmetical Properties of Finite Groups Wujie Shi (施武杰） Suzhou University, China ISCHIA GROUP THEORY 2010 April 2010, ITALY

  2. In memory of Maria Silvia Lucido (1963-2008), 2005 St Andrews

  3. 0. Introduction Let G be a finite group and Chi(G) be one of the following sets: Ch1(G) = |G|, that is, the order of G; Ch2(G) = e(G)= { o(g) | gG }, that is, the set of element orders of G, (spectrum); Ch3(G) = cs(G)= { |gG| | gG }, that is, the set of conjugacy class sizes of G; Ch4(G) = cd(G), that is, the set of irreducible character degrees of G. Our aim is to study the structure of G under certain arithmetical hypotheses of Chi(G), i = 1, 2,3 or 4. Except the above quantitative sets, we may define Ch5(G) be the set of the maximal subgroup orders of G, Ch6(G) be the set of Sylow normalizer orders of G, and the other quantitative sets.

  4. Ch1(G) = |G| related Sylow theorem, odd order theorem, and some famous results; Ch2(G) = e(G)= { o(g) | gG } related some works and conjectures of mine and my colleagues; Ch3(G) = cs(G)= { |gG| | gG } related J.G. Thompson’s conjecture; Ch4(G) = cd(G), related Huppert’s following conjecture: Let G be a group and M a simple group. If cd(G) = cd(M), then GMx A, where A is a finite abelian group. (B. Huppert, Illinois J. Math. 44(2000), 828-842.)

  5. Question AIf Chi(G) is given, what can we say about the structure of G? For the set Chi(G), i = 2, 3 or 4, we can define a graph (G) as follows: Its vertices are the primes dividing the numbers in Chi(G),and two distinct vertices p,q are connected if pq | m holds for m Chi(G). Question B If we know the information of graph i (G) i =2,3 or 4, what can we say about the structure of G?

  6. For any a Chi(G), i = 2, 3 or 4, for example, a Ch2(G), we define M2(a), the multiplicity ofa in G as the number of elements of order a. Also DC3,2(G)= |Ch3'(G)|-|Ch2(G)|, the difference of conjugacy classes number |Ch3'(G)| and same order classes number in G. Furthermore QC1,3(G)= |G| / |Ch3'(G)|, the quotient of |G| and the number of conjugacy classes. (Also see J. Algebra, 300(2006)) Question C If Ch(G) and M(a), for all aG, are known, what can we say about the structure of G? If DC(G) is "small", what can we say about the structure of G? If QC(G) is known what can we say about the structure of G? Question D Let bi(G) = max {Chi(G)}, i = 2, 3 or 4. If some information of bi(G) is known, what can we say about the structure of G?

  7. 1. Some Concepts and Theorems • Classification Theorem. • If G is a finite simple groups, then G is isomorphic • to one of the following groups: • Abelian simple groups Zp, p is prime; • 2. non-Abelian simple groups: • (a) alternating groups An, n  5, • (b) 16 infinite series groups of Lie type, 2F4(2)’, • (c) 26 sporadic groups.

  8. In Feb. 1981, the classification of the finite simple groups was called completed, representing one of the most remarkable achievements in the history of mathematics, and proved it in 2003 (the efforts of several hundred mathematicians over a period of 50 years, full proof covered something between 5000 and 10000 journal pages).

  9. Some famous results for |G| (Ch1(G)) Sylow(1832-1918) theorem Lagrange(1736-1813) theorem |G| is odd G solvable (1906, Burnside(1852-1927) posed, 1963, W. Feit(1930-2004) and J.G. Thompson(1932- ) proved, Full paper 254 pages, filled an entire issue of the Pacific Journal of Mathematics, 1963, Thompson got Fields prize for it.) paqb theorem: |G| = paqbG solvable Cauchy(1789-1857) theorem: p | |G| pe(G)

  10. Denote by e(G) (that is, Ch2(G) in this talk) the set of all orders of elements in G. For the set {|G|}, there are many famous and interesting results. But we do not know more information for the set e(G) . Obviously, e(G) is a subset of the set Z+of positive integers, and a difficult problem is: “ Which subset of Z+ constitute a set of orders of element of a group? ” If m  e(G) and n | m, we have ne(G), that is, it has a closure property for division. We only know it for this set.

  11. Some interesting results for e(G) If we divide the set e(G) into {1}, the set e’(G) consisting of primes and the set e”(G) consisting of composite numbers, then we have Theorem 1.1(Deng, Shi; 1997) Let G be any finite group. Then |e’(G)|  |e”(G)| + 3, and if the equality holds, then G is simple. Moreover, these simple groups are all determined only by the set e(G). Theorem 1.2(Brandl, Shi; 1991) Let G be a finite group whose element orders are consecutive integers. That is, e(G) = {1, 2, 3, … , n}. Then n  8.

  12. History ● In 1902, W. Burnside put forward the celebrated problem: Let G be a finitely generated group. If for every gG, the order o(g) of g is finite, is G necessarily finite? Although the problem of Burnside has been given the negative solution, it stresses the role of the orders of elements in the structure of groups. ● In 1937, B.H. Neumann discussed such groups G with e(G) = {1, 2, 3}. ● In 1957, G. Higman researched the finite groups G all of whose elements are of prime power order. that is, e(G) = {pn, qm, ……}, where p, q, … are primes. We call such groups finite EPPO-groups. In 1962, M. Suzuki, In 1981, R. Brandl researched the nonsolvable EPPO-groups. ● In 1982, we discussed finite EPO-group and EPPO-groups. An EPO-group, that is, e(G) = {1, p, q, ……}, and we got an interesting result. We may characterize A5 only using the element orders: Let G be a finite group. Then GA5e(G) = {1, 2, 3, 5}.

  13. 2. On the order of finite groups The order |G| of a group G is an integral property of groups. For the special number |G| we can determine the group G, for example, |G| = p, p is a prime. That is, there is only one group of order p (from the view of isomorphism) which is an Abelian simple group Zp. How many different groups with the given order exist? Many specialists of group theory researched the number f(n) of distinct groups G with the same order, as J.G. Thompson, G. Higman, C.C. Sims and others. We may refer to some references.

  14. Let G be a finite group. Denote |G| = n and f(n) the number of distinct groups G with |G| = n. The following table list the number of distinct groups of each order from 1 through 20: (M. Hall, The Theory of Groups, page 52.)

  15. Moreover, if assume G is simple, thenf simple(n) = 0, 1 or 2. That is, W. Kimmerle, R. Lyons, R. Sandling and D.N. Teague proved (1990):Theorem 2.1. Let G be a simple group and |G|=|M|, where M is a known finite simple group. Then one of the following conclusions holds.(1) If |M| = |A8| = |L3(4)|, then G  A8 or L3(4);(2) If |M| = |Bn(q)| = |Cn(q)| where n 3 and q is odd , then G  Bn(q) or Cn(q);(3) If |M| is not the above case (1) or (2), then G  M.

  16. Now we consider the inverse problem of above discussion. That is, we discuss the inverse function f 1, that is, for given k, which n satisfies f (n) = k ? It is easy to prove f (n) = 1 if and only if ( n, (n) ) = 1, where (n) is Euler function of n. Also the solutions for f (n) = 2, 3 and 4 is found. If 1 k  6000, it has proved there exist a number n such that f (n) = k . Conjecture 2.1(Guimin Wei;1998) For any k Z, there exist n, such that f(n) = k.

  17. 3. On orders of finite simple groups As early as 1940s R. Brauer and H.F. Tuan characterized some projective special linear groups using the conditions of the group order |G| and the simplicity of G. Theorem 3.1. Let G be a simple group of order prqb, where p, q and r are three different primes, and b is a positive integer. Then G  L2(5)or L2(7). Then many mathematicians characterized many simple groups from various point of view. For example, some mathematicians characterized the simple groups G using the conditions of |G| and the simplicity of G. Some Characterization using the decomposition of prime factors of |G| and the simplicity we may refer torelatedreferences.

  18. Definition 3.1. For any n Z, set (n) := { p | p prime, p | n }. For a finite group G, set (G):= (|G|). From paqb theorem we have, if G is simple, then |(G)|≥3. Using the number |(G)| M. Herzog got the following result. Theorem 3.2(M. Herzog; 1968). Let G be a finite simple group. If |(G)|=3, then G is isomorphic to one of the following groups: A5, L2(7), L2(8), A6, L2(17), L3(3), U3(3) or U4(2). D. Gorenstein called above eight simple groups as simple K3- groups (i.e. |(G)|=3). We determined all simple K4-groups (i.e. |(G)|=4 ) using the classification theorem, but we do not know the number of simple K4-groups is finite or infinite.

  19. Theorem 3.3(Shi; 1991) Let G be a simple K4-group. Then G is isomorphicto one of the following groups: An , n = 7, 8, 9, 10; M11, M12, J2; L2(q), q = 16, 25, 49, 81; L3(q), q = 4, 5, 7, 8, 17; L4(3); O5(q), q = 4, 5, 7, 9; O7(2), O8+(2), G2(3); U3(q), q = 4, 5, 7, 8, 9; U4(3); U5(2); 3D4(2); 2F4(2); Sz(8), Sz(32); and L2(r), r being prime and satisfying the following equation: r2 1 = 2a3buc, (1) where a  1, b  1, c  1, u prime, u > 3; L2(2m) and satisfying the following equations: where m 1, u, t primes, t > 3, b1; L2(3m) and satisfying the following equations: where m 1, u, t odd primes, c  1, b  1.

  20. Remark 3.1. Recently some authors investigate these Diophantine systems and proved that equations (2), (3) and (4) have no other solution except m = 5, u = 11, c = 2 in (3) when the exponents are greater than 1.(Y. Bugeaud, Z. Cao and M. Mignotte, On simple K4-groups, J. Algebra, 241(2001), 658~668.) Question 3.1. The number of simple K4-groups is determined by the number of solution of equations (1) ~ (4). But it is unknown whether the number of solution is finite or infinite. In other words, is the number of simple K4-groups finite or infinite? It was verified that the number of simple K4-groups is 101 if the largest prime divisor of the orders of groups is less than 1060. I believe that the number of simple K4-groups is infinite, but we do not prove it.

  21. 4. On the set of element orders Now we are interested in comparing groups by the set e(G) rather than by the orders of the groups themselves. Obviously, e(G) is a subset of the set Z+of positive integers, and it is more difficult problem which subset of Z+ can become the sets of element orders of groups. First, we give the following function h defined on subsets of positive integers. Definition 4.0. For a set of positive integers let h() be the number of isomorphism classes of finite group G such that e(G) =. For a given group G, we have h(e(G))  1. And a group G is called characterizable (or recognizable )if h(e(G)) = 1.

  22. In 1982, we studied such finite groups in which every non-identity element has prime order, i.e. finite EPO groups and got an interesting result: G A5  e(G) = {1, 2, 3, 5}, that is, h(e(A5 )) = 1 After this, we developed such characterizations for all finite simple groups using the "two orders“ (|G| and e(G) ). And for the finite nonsolvable groups only using the "set of element orders"(or "spectrum“, e(G)). That is, we researched the following characterizations of two kinds.

  23. Charactering all f.s.g using “two orders” Characterizing all finite simple groups unitization using only the two sets |G| (Ch1(G)) and e(G) (Ch2(G)). Now we give an outline of the following (posed in 1987) ( Question 12.39 in the Kourovka Notebook ): Theorem 4.1 Let G be a group and M a finite simple group. Then G  M if and only if (a) e(G)=e(M), and (b) |G| = |M|.

  24. Proof. Using the classification of finite simple groups (CFSG). Also, we depend on the following two Lemmas: Lemma 4.2 (K.W. Gruenberg, O. Kegel; 1975) If G is a finite group with its prime graph having more than one component, then G is one of the following groups: (a) a Frobenius or 2-Frobenius group; (b) a simple group; (c) an extension of a 1 –group by a simple group; (d) an extension of a simple group by a 1- solvable group; (e) an extension of a 1- group by a simple group by a 1- group. We answered the following E.Artin’s problem Using CFSG: Find out all simple groups of order |G| which Contain a Sylow subgroup of order greater than |G|1/3.

  25. Lemma 4.3 (Shi; 1992) Let G be a finite non-abelian simple group. If pk || |G|, where p is an odd prime, and |G|< p3k , then G is isomorphic to one of the following groups: (1) A simple group of Lie type in characteristic p; (2) A5 (p=5); A6 (p=3); A9 (p=3); (3) L2(p-1) (p is Fermat prime); L2(8) (p=3); U5(2)(p=3). For p = 2, we have a similar result. Corollary 4.4(G. Mendoza; 2008) For any prime p with p>3, The smallest nonabelian simple group of order divisible by p is L2(p). From 1987 to 2003, in refs. [1-7] the authors proved that Theorem 4.1 is correct for all finite simple groups except Bn, Cn and Dn (n even).

  26. In 2007, the authors of refs. [8, 9] proved that e(Bn(q)) ≠ e(Cn(q)) for all odd q. Recently, A. V. Vasilev, M. A. Grechkoseeva, and V. D. Mazurov proved that Theorem 4.1 is correct for finite simple Groups Bn, Cn and Dn (n even) (see refs. [10,11]). They analyze the problem of formerly proof (depend on Lemma 4.3) and established several lemmas. Thus we proved Theorem 4.1, and all finite simple groups can determined by their “two orders”.

  27. However, it does not true for some very small groups. Examples: 1. D8(dihedral group of order 8) and Q8(quaternion group) |D8| = |Q8| = 8 and e(D8) = e(Q8) = {1, 2, 4}, but D8 ≇ Q8 , they are different groups. • G1= Z3  Z3  Z3 and G2= (Z3  Z3) ⋊Z3 (semi-direct product), |G1| = |G2| and e(G1) = e(G2) = {1, 3}, but G1 and G2 are not isomorphism. Question 4.5 Let G1, G2,…, Gk be finite groups. If e(G1) = e(G2) = … = e(Gk) , whether or not the order of simple group Gi (if exsits) is the smallest? . If |G1|= |G2| = … = |Gk| , whether or not the number |e(Gi)| of simple group Gi (if exsits) is the smallest?

  28. 5. Many f.s.g. are characterizable using only the set e(G). Now we are interested in comparing groups by the set e(G) rather than by the orders of the groups themselves. Obviously, e(G) is a subset of the set Z+of positive integers. First, we recall the following function h defined on subsets of positive integers (Definition 4.0.). Definition 5.1. For a set of positive integers let h() be the number of isomorphism classes of finite group G such that e(G) =. For a given group G, we have h(e(G))  1. And a group G is called characterizable (or recognizable )if h(e(G)) = 1.

  29. Theorem 5.2 The following groups are characterizable (recognizable) groups: (a) Alternating groups An, where n = 5,16, p, p+1, p+2, and p≥ 7 is a prime; Symmetric group Sn, where n = 7, 9,11,12,13,14,19, 20, 23, 24. (b) Simple groups of Lie type L2(q), q  9, L3(2m), Ln(2m), n=2k > 8, F4(2m), U3(2m), m≥ 2, the simple groups of Suzuki-Ree type Sz(22m+1), 2G2(32m+1), 2F4(22m+1); S4(32m+1) (m>0), G2(3m); L3(7), L4(3), L5(3), L5(2), L6(2), L7(2), L8(2), U3(9), U3(11), U4(3), U6(2), S6(3), O8(2), O10(2), 3D4(2), G2(4), G2(5), F4(2), 2F4(2)', 2E6(2); and nonsolvable groups PGL2(pm), m > 1, pm  9, L2(9).23 (  M10), L3(4).21. …. (c) All sporadic simple groups except J2. A.Kh. Zhurtov and V.D. Mazurov extended the characterization of the above groups L2(2m), from finite groups to periodic groups. For the case of h(e(G))=  we have Theorem 5.3 If all the minimal normal subgroups of G are elementary (especially G is solvable), or G is one of the following: A6, A10, L3(3), U3(3), U3(5), U3(7), U4(2), U5(2); J2; S4(q) (q  32m+1 and m > 0). …. Then h(e(G))=  .

  30. In the case of k-recognized groups (i.e. h(e(G)) = k), we have found infinite pairs of 2-recognizable groups as follows: • L3(q), L3(q), where  is a graph automorphism of L3(q) of order 2, q = 5, 29, 41, or q  ± 2(mod 5) and (6, (q1)/2)=2; (b) L3(9), L3(9).21; (c) S6(2), O8+(2); (S6(2) < O8+(2)) (d) O7(3), O8+(3); (O7(3) < O8+(3)) (e) L6(3), L6(3) ,where  is a graph automorphism of L6(3) of order 2; (f) U4(5), U4(5) ,where  is a graph automorphism of U4(5) of order 2.

  31. In the following jointed paper (Huiwen Deng, M. S. Lucido, and Wujie Shi, The number of isomorphism classes of finite groups with given element orders, Algebra and Logic, 41 (2002).), we proved that: If G be a finite group with t(G) > 2, where t(G) is the number of connected components of prime graph of G, then h(e(G))= 1 or . Thus, we posed the following conjecture: For any group G, h(e(G))= 1, 2 or . Butthe above conjecture is not correct.

  32. A.V. Zavarnitsine(J. Group Theory, 2004) proved that: For any r > 0, h(e( ))=r+1, and these r+1 groups are , where  is a field automorphism of , k = 0, 1, 2, … , r. But the following problem is open: Problem 5.4. Whether or not there exist two section-free finite groups G1 and G2 such that e(G1) = e(G2) and h(e(G1)) is finite?

  33. 6. On Thompson’s first problem From 1987 to 1991, after I wrote some letters to Professor John G. Thompson and reported the above conjecture (Theorem 4.1). Thompson pointed that, “Good luck with your conjecture about simple groups. I hope you continue to work on it”, “I like your arguments”, “This would certainly be a nice theorem.”…, in his reply letters. Also, he posed the following two problems.

  34. For each finite group G and each integer d 1, let G(d) = xG | xd = 1. Definition 6.1.G1 and G2 are of the same order type if and only if |G1(d)|=|G2(d)|, d=1, 2, ..…. . Problem 6.2.(J.G. Thompson, 1987)Suppose G1 and G2 are groups of the same order type. Suppose also that G1 is solvable. Is it true that G2 is also necessarily solvable?

  35. In Thompson’s Private letter he pointed out that “The problem arose initially in the study of algebraic number fields, and is of considerable interest”.Also we know it stems from the need of research in differential geometry.

  36. In 1987, J.G. Thompson gave an example as follows: G1 = 24 : A7 and G2 = L3(4) : 22 both are maximal subgroups of M23. From this example we get G1 and G2 have the same "order equation" : |G1| = |G2| = n1 + n2 + n3 + n4 + n5 + n6 + n7 + n8 + n14 = 1 + 435 + 2240 + 6300 + 8064 + 6720 + 5760 + 5040 + 5760. e(G1) = e(G2) = {1, 2, 3, 4, 5, 6, 7, 8, 14}. It showed that the "order equation" can not decide the simple section of G. But if Thompson’s problem is affirmative, then the "order equation“ may determine the solvability of all finite groups. What is"order equation" ?

  37. In fact the groups of the same order type are such groups whose “order equations” are the same. That is, according to the element orders as the equivalent relation, the group G is partitioned into the same order classes. The “order equation” is |G| = n1 + n2 + … + ns , where nidenotes the number of elements of order i in G. Obviously the same order class is an union of several conjugacy classes and the partition of “order equation” is rougher than the class equation. If the conjugacy classes equation of G as follows |G| = c1 + c2 + … + ct. Thus we have t ≥ s. Then there exists a non-negative integer k such that k = t – s, and we call this group co(k)-group. Clearly, finite co(0)-groups is the groups posed by Syskin. W. Feit and G.M. Seitz, J. P. Zhang had proved well-known Syskin problem independently. The result of Syskin problem is that: If any elements of finite group with same order are conjugate, then GSi , i = 1, 2 or 3.

  38. We study the following two generalizations: (1) The case of t  s 2. (see Xianglin Du and Wujie Shi, Finite groups with conjugacy classes number one greater than its same order classes number (with), Comm. in Algebra, 34(2006), 1345-1359. ) (2) The elements of same order outside Z(G) are conjugate. (see Guohua Qian Wujie Shi and Xingzhong You, Conjugacy classes outside a normal subgroup, Science in China, Ser. A: Mathematics, 2007，50:10, 1493-1500.) Also, we used the classification of the finite simple groups in the discussion of the above two cases.

  39. If G1 and G2 are finite groups of the same order type, then e(G1) = e(G2) and |G1| = |G2|. Since Theorem 4.1 is proved, then we have the following corollary: Let G1 and G2 be finite groups of the same order type. If G1 is solvable, then G2 is not simple. I think the main difficulty is that the induction is not useable for thus groups in discussion. Dr. M. Xu proved that if G1 and G2 are of the same order type, and G1 possess Sylow tower (or G1 is supersolvable), then G2 is solvable. Lastly, Rulin Shen and Shi proved: Let G1 be a solvable group whose prime graph(G1) is not connection. If G1 and G2 are the groups of same order type, then G2 is also solvable.

  40. 7. On Thompson’s second problem Another Thompson’s conjecture, which is also aim at characterizing all finite simple groups by a quantity set, is posed in 1988, which is appeared in another communication letter: If G is a finite group, set N(G) = Ch3(G) = n Z+ | G has a conjugacy class C with |C| = n 7.1 Conjecture (J.G. Thompson)If G and M are finite groups and N(G) = N(M), and if in addition, M is a non- Abelian simple group while the center of G is 1, then G and M are isomorphic.

  41. On the above conjecture , Dr. G.Y. Chen has dealt with the situation that M is a finite simple group having a non- connected prime graph in his Ph.D. thesis, and published the case of t(M)  3, where t(M) is the number of connected components of prime graphof M. 7.2. QuestionFor Thompson’s conjecture, how deal with the case of connected prime graph？ Now, some small simple groups, as A10, L4(4) ( A. V. Vasilev, Siberian Electronic Mathematical Reports, 2009), whose prime graph is connected, have been proved for Thompson’s conjecture.

  42. References • Wujie Shi, A new characterization of the sporadic simple groups, Group Theory - Porc. Singapore Group Theory Conf. 1987, Walter de Gruyter Berlin-New York, 1989. • Wujie Shi and Jianxing Bi, A characteristic property for each finite projective special linear group (with J.X. Bi), Lecture Notes in Math., Springer-Verlag, 1456(1990). • Wujie Shi and Jianxing Bi, A characterization of Suzuki–Ree groups, Sci. in China, Ser. A, 34 (1991). • Wujie Shi and Jianxing Bi, A new characterization of the alternating groups, Southeast Asian Bull. Math., 16 (1992). • Wujie Shi, The pure quantitative characterization of finite simple groups (I), Prog. Nat. Sci., 4 (1994). • Hongping. Cao and Wujie Shi, Pure quantitative characterization of finite projective special unitary groups, Sci. China, Ser. A, 45 (2002). 7. Mingchun Xu and W. Shi, Pure quantitative characterization of finite simple groups 2Dn(q) and Dl(q) (l odd), Alg. Coll., 10 (2003). • Wujie Shi, Pure quantitative characterization of finite simple groups, Front. Math. China, 2(2007). 9. M. A. Grechkoseeva, On difference between the spectra of the simple groups Bn(q) and Cn(q), Siberian Mathematical Journal, 48 (2007).

  43. 10. A. V. Vasilev, M. A. Grechkoseeva, and V. D. Mazurov, On finite groups isospectral to simple symplectic and orthogonal groups, Siberian Math. J., 50( 2009). 11. A. V. Vasilev, M. A. Grechkoseeva, and V. D. Mazurov, Characterization of the finite simple groups by spectrum and order, Algebra and Logic, 48( 2009). The End Thank You ! 谢谢！

More Related