Mathematics and Computer Science
Volume 1, Issue 1, May 2016, Pages: 17-20

Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree

Liguo He*, Yaping Liu, Jianwei Lu

Dept. of Math., Shenyang University of Technology, Shenyang, PR China

Email address:

(Liguo He)

*Corresponding author

To cite this article:

Liguo He, Yaping Liu, Jianwei Lu. Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree. Mathematics and Computer Science. Vol. 1, No. 1, 2016, pp. 17-20. doi: 10.11648/j.mcs.20160101.14

Received: April 11, 2016; Accepted: May 3, 2016; Published: May 28, 2016


Abstract: For a finite group , we write  to denote the prime divisor set of the various conjugacy class lengths of  and  the maximum number of distinct prime divisors of a single conjugacy class length of . It is a famous open problem that  can be bounded by . Let G be an almost simple group  such that the graph  built on element orders is a tree. By using Lucido’s classification theorem, we prove  except possibly when  is isomorphic to , where  is an odd prime and  is a field automorphism of odd prime order . In the exceptional case, . Combining with our known result, we also prove that for a finite group  with  a forest, the inequality  is true.

Keywords: Prime Graph, Conjugacy Class Length, Almost Simple Group


1. Introduction

Throughout the paper, we only consider finite groups. For a finite group , write  to denote the prime divisor set of the various conjugacy class lengths of  and  the maximum number of different prime divisors of a single conjugacy class length of . And  stands for prime divisor set of the positive integer  and write  for , here  denotes the order of . Considering some similarities between the influence of character degrees and conjugacy class lengths on groups, in 1989, Huppert once asked [1] whether the inequality  holds for every solvable group. It was shown in [2, 4, 6] that this is true when  is at most 3. Specifically, the case  is proven in [4], the case  for the solvable  is in [6], and the cases  for the nonsolvable  and  are finished in [2], respectively. For solvable groups, it is proved  in [7], and an improved version  in [20]. Generally, Casolo proved in [1, Corollary 2] that the inequality is true except possibly when  is  nilpotent with abelian Sylow  subgroups for at least two prime divisors of . And yet Casolo and Dolfi in [3, Example 2] show the inequality is invalid by constructing an infinite family of group examples . Specifically, the quotient  approaches  (from below) as  approaches infinity. The prime numberdivides each . When take , we may obtain an infinity family of counterexamples such that each  is metabelian and super solvable and the constant is 3 in that inequality. Note that the subscripts  in these counterexamples are sufficient large. By observation of these counterexamples, in 1998, Zhang further conjectured [20] the weak version of that inequality should hold by saying that "Now, it seems true that  for any finite solvable group and probably also for any finite group." We attach a prime graph  to a finite group : its vertices are , and any two vertices are adjacent by an edge just when  has an element of order . We use  to denote the connected components of the graph , in particular,  if  is of even order. Furthermore, the graph  is called a tree when it is a connected graph without any loop; and  is called a forest when each connected component is a tree. In this note, we prove the following results.

Theorem A. Suppose that  is an almost simple group such that  is a tree. Then  except possibly when  is isomorphic to , whereis an odd prim and  is a field automorphism of odd prime order . In the exceptional case, .

Theorem B. Suppose that  is a finite group such that is a forest. Then .

In the proof of Theorem A, we apply the classification result due to Lucido (Theorem 2.2). In the process, GAP [6] plays a crucial role. In essence, we use finite simpl egroup classification theorem.

Unless otherwise specified, we adopt the standard notation and terminology as presented in [9].

2. Preliminaries

The following fact is useful and basic on conjugacy class lengths, which will be frequently applied without reference.

Lemma 2.1 ((Lemma 33.2 of [9])). Letbe a normal subgroup of  and . Then

1.  divides  for any , and so.

2.  divides  for any , and thus .

Theorem 2.2. If  is an almost simple group with  a tree, then  is one of the following types of groups.

1.  for the alternating group  of degree 6.

2.  such that  is a prime,  is an odd prime and  is a field automorphism of order .

3. .

4. .

5. , where  is a diagonal automorphism of order 2.

6.  with  is a graph-field automorphism of order 2.

7. , with  and  a field automorphism of order 2.

8. .

 with  a field automorphism of order . Here  is an odd prime.

Proof. This is Lemma 3 of [11].

Lemma 2.3. Assume that  is a finite group with disconnected . Then the inequality  is true.

Proof. This is Theorem A of [8].

Lemma 2.4. Let  be a finite group with . Then .

Proof. By [2, 3, 5], the inequality is valid when . Otherwise, we see  and so , as wanted.

Theorem 2.5. Let  be a finite group such that  is a tree, then .

Proof. This is Theorem 6 of [11].

Lemma 2.6. Let  be an almost simple group over the nonabelian simple group , then .

Proof. It is known  by Theorem 33.4 of [9]. If  has a maximal central Hall subgroup , then we can write . As  is a simple group, the intersection  is trivial. It follows that  divides , and so . We obtain  acts trivially on , however, this is a contradiction since . Therefore  is trivial an .

3. Main Results

Theorem 3.1. Suppose that  is an almost simple group such that  is a tree. Then  except possibly whenis isomorphic to , where is an odd prime and is a field automorphism of odd prime order. In the exceptional case, .

Proof. We apply Theorem 2.2 above to prove the claims.

If  is isomorphic to , then we know via GAP [6] that  is trivial and , thus we obtain , as desired.

Assume thatis isomorphic to . Here is an odd prime and  is a field automorphism of order  which is also an odd prime. By Lemma 4.1of [10], we know  is trivial and so . It is known when  is an odd prime. As in [4], denote by  an element of order  in  and by  the element of order 2 in the centre of . Then it is known that the class size of  in  is  (which is the conjugacy class size corresponding to , and so we conclude that  for odd . The field automorphism  indeed leave the class of  invariant by [10, Lemma 4.1]. We further get

for odd . Note that is an odd prime.

Assume now that . Then we see

and

,

By using GAP [6], we know that  has 43 conjugacy class lengths,

 and ; and has 56 conjugacy class lengths,  and . Hence we obtain , as required.

If  is isomorphic to , then the application of GAP yields that  has twenty conjugacy class lengths,  and , and so , as wanted.

Next, consider the case that, where  is a diagonal automorphism of order 2. Using GAP, we reach that  has 20conjugacy class lengths,  and . Its automorphism group  has 61 conjugacy class lengths,  and . As  is an almost simple group, we achieve that  and , thus , as desired.

Now, suppose that  with  is a graph-field automorphism of order 2. As  is of order 2, we attain  is either  or else

. When , by GAP, we know and , and so. When , we also get that  because  is of order 2 which is in .

The next case is , with  and  a field automorphism of order 2. For , we get via GAP that  and . Also since

 this yields . For , by GAP, we attain that  and . Also , this implies .

For , we get by GAP that and . Also , this shows , as desired.

Assume that . By using GAP, it follows that  and . Note that  is an almost simple group with , thus .

Assume finally that  is isomorphic to , where  is a field automorphism of odd prime order . Let be a Sylow 2-subgroup of .

By Proposition 1 of [12], we see  for any nontrivial element .

Using Lemmas 1 and 2 of [12], we may pick the noncentral element , which is indeed a specific form of  (by [12, Theorem 7]). Here neither of  or  can equal 0, 1. Then the centralizer  is properly contained in , and so . If  belongs to , then . Otherwise,  is a Sylow -subgroup. By [7, Theorem 9], we have  and so . The whole proof is complete.

Some remarks on  are made here. By [4], we see that  for the odd  and  for .

Lemma 2 of [11] yields  unless. Some direct calculations by GAP show that ,  and  are all equal to 3, but  and  are 2. Observe that   and , moreover .

Theorem 3.2. Suppose that  is a finite group such that is a forest. Then

.

Proof. If  is disconnected, then Lemma 2.3 yields the result. Otherwise,  is a tree, and so  (by Lemma 2.5). Applying Lemma 2.4, we get the result.

In this note, we show Huppert’s problem has affirmative answer for the finite group whose prime graph a forest, and even has better result when the group is an almost simple group with prime graph a tree.

Acknowledgements

Project supported by NSF of China (No. 11471054) and NSF of Liaoning Education Department (No. 2014399).


References

  1. C. Casolo, Prime divisors of conjugacy class lengths in finite groups, Rend. Mat. Acc. Lincei, 1991, Ser. 9, 2: 111-113.
  2. C. Casolo, Finite groups with small conjugacy classes, Manuscr. Math., 1994, 82: 171-189.
  3. D. Chillig, M. Herzog, On the length of conjugacy classes of finite groups, J. Algebra, 1990, 131: 110-125.
  4. L. Dornhoff, Group representation theory, Part A: Ordinary representation theory, Marcel Dekker, New York, 1971.
  5. P. Ferguson, Connections between prime divisors of the conjugacy classes and prime divisors of , J. Algebra, 1991, 143: 25-28.
  6. The GAP Group, GAP- Groups, algorithms, and programming, version 4.6,http://www.gap-system.org, 2013.
  7. D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc. 276, 1983 (vol.42).
  8. L. He, Y. Dong, Conjugacy class lengths of finite groups with disconnected prime graph, Int. J. Algebra, 2015, 9(5): 239 - 243.
  9. B. Huppert, Character theory of finite groups, DeGruyter Expositions in Mathematics 25, Walter de Gruyter & Co.: Berlin. New York, 1998.
  10. M. L. Lewis and D. L. White, Nonsolvable groups with no prime dividing three character degrees, J. Algebra, 2011, 336: 158-183.
  11. M. S. Lucido, Groups in which the prime graph is a tree, Bollettino U.M.I., 2002, Ser. 8, 5-B: 131-148.
  12. M. Suzuki, On a class of doubly transitive groups, Ann. Math., 1962, 75:105-145.
  13. J. P. Zhang, On the lengths of conjugacy classes, Comm. Algebra, 1998, 26(8): 2395-2400.

Article Tools
  Abstract
  PDF(488K)
Follow on us
ADDRESS
Science Publishing Group
548 FASHION AVENUE
NEW YORK, NY 10018
U.S.A.
Tel: (001)347-688-8931