A characterization of some alternating groups A p+8 of degree p + 8 by OD

SpringerPlus, Jul 2016

Let \(A_n\) be an alternating group of degree n. We know that \(A_{10}\) is 2-fold OD-characterizable and \(A_{125}\) is 6-fold OD-characterizable. In this note, we first show that \(A_{189}\) and \(A_{147}\) are 14-fold and 7-fold OD-characterizable, respectively, and second show that certain groups \(A_{p+8}\) with that \(\pi ((p+8)!)=\pi (p!)\) and \(p<1000\), are OD-characterizable. The first gives a negative answer to Open Problem of Kogani-Moghaddam and Moghaddamfar.

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A characterization of some alternating groups A p+8 of degree p + 8 by OD

A characterization of some alternating groups A of degree p   8 by OD p+8 + Shitian Liu 1 Zhanghua Zhang 0 0 Sichuan Water Conservancy Vocational College , Chongzhou, Chengdu 643000, Sichuan , People's Republic of China 1 School of Science, Sichuan University of Science and Engineering , Xueyuan Street, Zigong 643000, Sichuan , People's Republic of China Let An be an alternating group of degree n. We know that A10 is 2-fold OD-characterizable and A125 is 6-fold OD-characterizable. In this note, we first show that A189 and A147 are 14-fold and 7-fold OD-characterizable, respectively, and second show that certain groups Ap+8 with that π((p + 8)!) = π(p!) and p < 1000, are OD-characterizable. The first gives a negative answer to Open Problem of Kogani-Moghaddam and Moghaddamfar. Definition 2 (Moghaddamfar et al. 2005) A finite group M is called k-fold OD-charac terizable if hOD(M) = k. Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Up to now, some groups are proved to be k-fold OD-characterizable and we can refer to the corresponding references of Akbari and Moghaddamfar (2015) . Concerning the alternating group G with s(G) = 1, what’s the influence of OD on the structure of group? Recently, the following results are given. Theorem 3 The following statements hold: (1) The alternating group A10 is 2-fold OD-characterizable (see Moghaddamfar and Zokayi 2010). (2) The alternating group A125 is 6-fold OD-characterizable (see Liu and Zhang Submitted). (3) The alternating group Ap+3 except A10 is OD-characterizable (see Hoseini and Moghaddamfar 2010; Kogani-Moghaddam and Moghaddamfar 2012; Liu 2015; Moghaddamfar and Rahbariyan 2011; Moghaddamfar and Zokayi 2009; Yan and Chen 2012; Yan et al. 2013; Zhang and Shi 2008; Mahmoudifar and Khosravi 2015) . (4) All alternating groups Ap+5, where p + 4 is a composite and p + 6 is a prime and 5 = p ∈ π(1000!), are OD-characterizable (see Yan et al. 2015) . In Moghaddamfar (2015), A189 is at least 14-fold OD-characterizable. In this paper, we show the results as follows. Theorem 4 The following hold: (1) The alternating group A189 of degree 189 is 14-fold OD-characterizable. (2) The alternating group A147 of degree 147 is 7-fold OD-characterizable. These results give negative answers to the Open Problem (Kogani-Moghaddam and Moghaddamfar 2012). Open Problem (Kogani-Moghaddam and Moghaddamfar 2012) All alternating groups Am, with m = 10, are OD-characterizable. We also prove that some alternating groups OD-characterizable. Ap+8 with p < 1000 are Theorem 5 Assume that p is a prime satisfying the following three conditions: (1) p = 139 and p = 181, (2) π((p + 8)!) = π(p!), (3) p ≤ 997. Let G be a finite group, then let Soc(G) denote the socle of G regarded as a subgroup which is generated by the minimal normal subgroup of G. Let Sylp(G) be the set of all Sylow p-subgroups Gp of G, where p ∈ π(G). Let Aut(G) and Out(G) be the automorphism and outer-automorphism group of G, respectively. Let Sn denote the symmetric groups of degree n. Let p be a prime divisor of a positive integer n, then the p-part of n is denoted by np, namely, np n. The other symbols are standard (see Conway et al. 1985, for instance) . Some preliminary results In this section, some preliminary results are given to prove the main theorem. Lemma 6 Let S = P1 × · · · × Pr, where Pi’s are isomorphic non-abelian simple groups. Then Aut(S) = Aut(P1) × · · · × Aut(Pr ).Sr. Proof See Zavarnitsin (2000). Lemma 7 Let An (or Sn) be an alternating (or a symmetric group) of degree n. Then the following hold. (1) Let p, q ∈ π(An) be odd primes. Then p ∼ q if and only if p + q ≤ n. (2) Let p ∈ π(An) be odd prime. Then 2 ∼ p if and only if p + 4 ≤ n. (3) Let p, q ∈ π(Sn). Then p ∼ q if and only if p + q ≤ n. Proof It is easy to get from Zavarnitsin and Mazurov (1999). Lemma 8 The number of groups of order 189 is 13. Proof See Western (1898). Lemma 9 Let P be a finite simple group and assume that r is the largest prime divi sor of |P| with 50 < r < 1000. Then for every prime number s satisfying the inequality (r − 1)/2 < s ≤ r, the order of the factor group Aut(P)/P is not divisible by s. Proof It is easy to check this results by Conway et al. (1985) and Zavarnitsine (2009). Let n = p1α1 p2α2 · · · prαr where p1, p2, . . . , pr are different primes and α1, α2, . . . , αr are positive integers, then exp(n, pi) = αi with piαi | n but piαi+1 ∤ n. Lemma 10 Let L := Ap+8 be an alternating group of degree p + 8 with that p is a prime and π(p + 8)! = π(p!). Let |π(Ap+8)| = d with d a positive integer. Then the following hold: (1) deg(p) = 4 and deg(r) = d − 1 for r ∈ {2, 3, 5, 7}. (2) exp(|L|, 2) ≤ p + 7. (3) exp(|L|, r) = i∞=1[ pr+i8 ] for each r ∈ π(L)\{2}. Furthermore, exp(|L|, r) < p +28 where 5 ≤ r ∈ π(L). In particular, if r > [ p +28 ], then exp(|L|, r) = 1. Proof By Lemma 7, it is easy to compute that for odd prime r, p · r ∈ ω(L) if an (...truncated)


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Shitian Liu, Zhanghua Zhang. A characterization of some alternating groups A p+8 of degree p + 8 by OD, SpringerPlus, 2016, pp. 1128, Volume 5, Issue 1, DOI: 10.1186/s40064-016-2763-7