Research Article | Open Access
Shitian Liu, Yong Yang, "On Thompson’s Conjecture for Alternating Groups ", The Scientific World Journal, vol. 2014, Article ID 752598, 10 pages, 2014. https://doi.org/10.1155/2014/752598
On Thompson’s Conjecture for Alternating Groups
Let be a group. Denote by the set of prime divisors of . Let be the graph with vertex set such that two primes and in are joined by an edge if has an element of order . We set to denote the number of connected components of the prime graph . Denote by the set of nonidentity orders of conjugacy classes of elements in . Alavi and Daneshkhah proved that the groups, where with , are characterized by . As a development of these topics, we will prove that if is a finite group with trivial center and with composite, then is isomorphic to .
In this paper, all groups considered are finite and simple groups are nonabelian simple groups. Let be a finite group and let be its center. For any , suppose that denotes the conjugacy classes in containing and denotes the centralizer of in . We will use to denote the set has a conjugacy class of size . Thompson in 1987 gave the following conjecture with respect to .
Thompson’s Conjecture (see , Question 12.38). If is a finite simple nonabelian group, is a finite group with trivial center, and , then .
Let denote the set of all prime divisors of . Let be the graph with vertex set such that two primes and in are joined by an edge if has an element of order . We set to denote the number of connected components of the prime graph . A classification of all finite simple groups with disconnected prime graph was obtained in [2, 3]. Based on these results, Thompson’s conjecture was proved valid for many finite simple groups with (see [4, 5]). In Ahanjideh and Xu’s works, the groups , , , , , and are true for Thompson’s conjecture (see [6–10]). Alavi and Daneshkhah proved that the groups with , , and are characterized by (see ). So is there a group with connected prime graph for which Thompson’s conjecture would be true? Recently, the groups , , , and were proved valid for this conjecture (see [12–15]). As a development of these topics, we will prove that Thompson’s conjecture is true for the alternating groups of degree with composite.
We will introduce some notations used in the proof of the main theorem. Let be a group and a prime. Then denote by the Sylow -subgroup of . Let and denote the automorphism and outer automorphism groups of , respectively. Let denote the set of element orders of . The other notations are standard (see , for instance).
2. Preliminary Results
In this section we will give some preliminary results.
Lemma 1. Let , , and . Then (1);(2) divides ;(3)if , then .
Lemma 2. If and are finite groups with trivial centers and , then .
Proof. See [12, Lemma 3].
Lemma 3. Suppose that is a finite group with trivial center and is a prime from such that does not divide for all in . Then a Sylow -subgroup of is elementary abelian.
Proof. See [12, Lemma 4].
Lemma 4. Let be a normal subgroup of , and . (1)If is the image of an element of in , then divides .(2)If , then .(3)If , then divides .
Proof. See [12, Lemma 5].
Let denote the nonnegative integer such that but .
Lemma 5. Let be the alternating group of degree , where is a prime. Then the following hold.(1); in particular, .(2) for each . Furthermore, , where . In particular, if , then .
Proof. (1) By the definition of Gaussian integer function, we have that
(2) Similarly, as (1), we have that for an odd prime .
If , .
The proof is completed.
Let be the symmetric group of degree . Assume that the cycle has 1-cycles, 2-cycles, and so on, up to -cycles, where . Then the number of conjugacy classes in is Let be the alternating group of degree .
Lemma 6. Let . Then, for the size of the conjugacy class of in , one has the following. (1)If, for all even , and, for all odd , , then .(2)In all other cases, .In particular, .
Proof. See .
Lemma 7. If is a natural number, then there are at least prime numbers such that . Here(i) for ;(ii) for ;(iii) for ;(iv) for ;(v) for ;(vi) for .In particular, for every natural number , there exists a prime such that , and, for every natural number , there exists an odd prime number such that .
Proof. See Lemma 1 of .
Lemma 8. Let , and be positive integers such that . Then there exists a prime with the following properties:(i) divides ,(ii) does not divide for all ,with the following exceptions: , ; and ; .
Proof. See .
Lemma 9. With the exceptions of the relations and every solution of the equation has exponents ; that is, it comes from a unit of the quadratic field for which the coefficients and are primes.
Let be a nonabelian simple group and let O denote the order of the outer automorphism group of .
Proof. See .
3. Main Theorem and Its Proof
In this section, we give the main theorem and its proof.
Theorem 11. Let be a finite group with trivial center and with composite. Then is isomorphic to .
We divide the proof into the following lemmas.
Lemma 12. Let . Then the following hold.(1)If , then we can write with and conjugacy class sizes of -elements of are for possible with . In particular, if is an odd prime divisor of , then conjugacy class sizes of -element of are where and .(2)If , then one can write with and conjugacy class sizes of -elements of are for possible with .(3)If , then one can write with and conjugacy class sizes of -elements of are In particular, if , then the conjugacy class size of -elements of is (4)The following numbers from are maximality with respect to divisibility.(a)One of the following holds: if ; if ; if and with ;(b).
Lemma 13. Let be a finite group with trivial center and . Then and .
Proof. Note that . Since , every member from divides the order of and . So by Lemma 12, we have that .
Lemma 14. Suppose that is a finite group with trivial center and . Then the following hold.(1)There exist different primes from such that . In particular, the Sylow -subgroup of is a cyclic group of order where . There does not exist an element of order , , or .(2)For all , if is divisible at most by , then the Sylow -subgroup of is of order .
Proof. (1) By Lemma 7, there exist different prime numbers from such that .
From Lemmas 12 and 13, we have that the primes are prime divisors of and do not divide for all . Then by Lemma 3, is elementary abelian. Therefore if , then is an -number.
Let . Consider an element of with or by Lemma 12.
Assume that . Let be an element of having order . Then , , and by Lemma 1. Since is abelian, . Hence, . It follows that equals or by Lemma 12.
If equals , then . On the other hand, ; then we have that or Obviously, there is no number from such that and .
Therefore equals . In the following, we will consider the following three cases.
Case 1. if .
Obviously, . Therefore , a contradiction, since , , and the maximality of .
Case 2. if .
Obviously, . Therefore . Also we get a contradiction as in Case 1.
Case 3. if and with .
In this case, . It follows that . By Lemma 1, and so , a contradiction.
Assume that . Let . Since is elementary abelian, the numbers and are coprime. Let Then and . Therefore, or
On the other hand, the element of is of order . Since the Sylow -subgroup of is elementary abelian, then . It follows that by Lemma 12.
If , then , a contradiction. Hence . We consider the following three cases.