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)