Open Access

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

Contributed equally
SpringerPlus20165:1128

https://doi.org/10.1186/s40064-016-2763-7

Received: 12 April 2016

Accepted: 5 July 2016

Published: 19 July 2016

Abstract

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.

Keywords

Element orderAlternating groupSimple groupSymmetric groupDegree patternPrime graph

Mathematics Subject Classification

20D0520D0620D60

Background

For a group, it means finite, and for a simple group, it is non-abelian. If G is a group, then the set of element orders of G is denoted by \(\omega (G)\) and the set of prime divisors of G is denoted by \(\pi (G)\). Related to the set \(\omega (G)\) a graph is named a prime graph of G, which is written by GK(G). The vertex set of GK(G) is written by \(\pi (G)\), and for different primes pq, there is an edge between the two vertices pq if \(p\cdot q\in \omega (G)\), which is written by \(p\sim q\). We let s(G) denote the number of connected components of the prime graph GK(G).

Moghaddamfar et al in 2005 gave the following notions which inspire some authors’ attention.

Definition 1

(Moghaddamfar et al. 2005) Let G be a finite group and \(|G|=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}\), where \(p_{i}\)s are primes and \(\alpha _{i}\)s are positive integers. For \(p\in \pi (G)\), let \(\deg (p):=|\{q\in \pi (G)|p\sim q\}|\), which we call the degree of p. We also define \(D(G):=(\deg (p_{1}),\deg (p_{2}),\ldots ,\deg (p_{k}))\), where \(p_{1}<p_{2}<\cdots <p_{k}\). We call D(G) the degree pattern of G.

For a given finite group M, write \(h_{OD}(M)\) to denote the number of isomorphism classes of finite groups G such that (1) \(|G|=|M|\) and (2) \(D(G)=D(M)\).

Definition 2

(Moghaddamfar et al. 2005) A finite group M is called k-fold OD-characterizable if \(h_{OD}(M)=k\). 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. (1)

    The alternating group \(A_{10}\) is 2-fold OD-characterizable (see Moghaddamfar and Zokayi 2010).

     
  2. (2)

    The alternating group \(A_{125}\) is 6-fold OD-characterizable (see Liu and Zhang Submitted).

     
  3. (3)

    The alternating group \(A_{p+3}\) except \(A_{10}\) 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. (4)

    All alternating groups \(A_{p+5}\), where \(p+4\) is a composite and \(p+6\) is a prime and \(5\ne p\in \pi (1000!)\), are OD-characterizable (see Yan et al. 2015).

     

In Moghaddamfar (2015), \(A_{189}\) is at least 14-fold OD-characterizable. In this paper, we show the results as follows.

Theorem 4

The following hold:
  1. (1)

    The alternating group \(A_{189}\) of degree 189 is 14-fold OD-characterizable.

     
  2. (2)

    The alternating group \(A_{147}\) 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 \(A_m\), with \(m \ne 10\), are OD-characterizable.

We also prove that some alternating groups \(A_{p+8}\) with \(p<1000\) are OD-characterizable.

Theorem 5

Assume that p is a prime satisfying the following three conditions:
  1. (1)

    \(p\ne 139\) and \(p\ne 181\),

     
  2. (2)

    \(\pi ((p+8)!)=\pi (p!)\),

     
  3. (3)

    \(p\le 997\).

     
Then the alternating group \(A_{p+8}\) of degree \(p+8\) is OD-characterizable.

Let G be a finite group, then let \(\mathrm {Soc}(G)\) denote the socle of G regarded as a subgroup which is generated by the minimal normal subgroup of G. Let \(\mathrm {Syl}_{p}(G)\) be the set of all Sylow p-subgroups \(G_p\) of G, where \(p\in \pi (G)\). Let \(\mathrm {Aut}(G)\) and \(\mathrm {Out}(G)\) be the automorphism and outer-automorphism group of G, respectively. Let \(S_{n}\) 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 \(n_p\), namely, \(n_p\Vert n\). 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 \(S=P_1\times \cdots \times P_r\), where \(P_i\)s are isomorphic non-abelian simple groups. Then \(\mathrm {Aut}(S)=\mathrm {Aut}(P_1)\times \cdots \times \mathrm {Aut}(P_r).S_r\).

Proof

See Zavarnitsin (2000). \(\square\)

Lemma 7

Let \(A_{n}\) (or \(S_{n}\)) be an alternating (or a symmetric group) of degree n. Then the following hold.
  1. (1)

    Let \(p,q\in \pi (A_{n})\) be odd primes. Then \(p\sim q\) if and only if \(p+q\le n\).

     
  2. (2)

    Let \(p\in \pi (A_{n})\) be odd prime. Then \(2\sim p\) if and only if \(p+4\le n\).

     
  3. (3)

    Let \(p,q\in \pi (S_{n})\). Then \(p\sim q\) if and only if \(p+q\le n\).

     

Proof

It is easy to get from Zavarnitsin and Mazurov (1999). \(\square\)

Lemma 8

The number of groups of order 189 is 13.

Proof

See Western (1898). \(\square\)

Lemma 9

Let P be a finite simple group and assume that r is the largest prime divisor of |P| with \(50< r<1000\). Then for every prime number s satisfying the inequality \((r - 1)/2 < s \le r\), the order of the factor group \(\mathrm {Aut}(P)/P\) is not divisible by s.

Proof

It is easy to check this results by Conway et al. (1985) and Zavarnitsine (2009). \(\square\)

Let \(n=p_1^{\alpha _1}p_2^{\alpha _2}\cdots p_r^{\alpha _r}\) where \(p_1,p_2,\ldots , p_r\) are different primes and \(\alpha _1,\alpha _2,\ldots ,\alpha _r\) are positive integers, then \(\exp (n,p_i)=\alpha _i\) with \(p_{i}^{\alpha _i}\mid n\) but \(p_i^{\alpha _i+1}\nmid n\).

Lemma 10

Let \(L:=A_{p+8}\) be an alternating group of degree \(p+8\) with that p is a prime and \(\pi (p+8)!=\pi (p!)\). Let \(|\pi (A_{p+8})|=d\) with d a positive integer. Then the following hold:
  1. (1)

    \(\deg (p)=4\) and \(\deg (r)=d-1\) for \(r\in \{2,3,5,7\}\).

     
  2. (2)

    \(\exp (|L|,2)\le p+7\).

     
  3. (3)

    \(\exp (|L|,r)=\sum \nolimits _{i=1}^{\infty }[\frac{p+8}{r^i}]\) for each \(r\in \pi (L)\backslash \{2\}\). Furthermore, \(\exp (|L|,r)<\frac{p+8}{2}\) where \(5\le r\in \pi (L)\). In particular, if \(r>[\frac{p+8}{2}]\), then \(\exp (|L|,r)=1\).

     

Proof

By Lemma 7, it is easy to compute that for odd prime \(r, p\cdot r\in \omega (L)\) if and only if \(p+r\le p+8\). Hence \(r=3,5,7\). If \(r=2\), then since \(p+4\le p+8\), then \(2\cdot p\in \omega (L)\). This completes (1).

By Gaussian’s integer function,
$$\begin{aligned} \exp (|L|,2)&= \sum \limits _{i=1}^{\infty }\left[ \frac{p+8}{2^i}\right] -1\\&= \left( \left[ \frac{p+8}{2}\right] +\frac{p+8}{2^2}+ \left[ \frac{p+8}{2^3}\right] +\cdots \right) -1\\&\le \left( \frac{p+8}{2}+\frac{p+8}{2^2}+\frac{p+8}{2^3}+\cdots \right) -1\\&= p+7. \end{aligned}$$
This proves (2). Similarly, we can get (3). \(\square\)

Lemma 11

Let am be positive integers. If \((a,m)=1\), then the equation \(a^x\equiv 1\)(mod m) has solutions. In particular, if the order of a modulo m is h(a), then h(a) divides \(\phi (m)\) where \(\phi (m)\) denotes the Euler’s function of m.

Proof

See Theorem 8.12 of Burton (2002).\(\square\)

Lemma 12

Let p be a prime and \(L:=A_{p+8}\) be the alternating group of degree \(p+8\) with that \(\pi ((p+8)!)=\pi (p!)\). Given \(P\in \mathrm {Syl}_p(L)\) and \(Q\in \mathrm {Syl}_q(L)\) with \(11\le q<p\le 1000\). Then the following results hold:
  1. (1)

    The order of \(N_L(P)\) is not divisible by \(q^{s(q)}\), where \(s(q)=\exp (|L|,q)\).

     
  2. (2)

    If \(p\in \{113, 139, 199, 211, 241, 283, 293, 337, 467, 509, 619, 787, 797, 839, 863, 887, 953, 997\}\), then \(|N_L(Q)|\) is not divisible by p.

     
  3. (3)

    If \(p\in \{181, 317, 409, 421, 523, 547, 577, 631, 661, 691, 709, 811, 829, 919\}\), then there is at least a prime r with that the order of r modulo p is less than \(p-1\), where \(11\le r<p\) and \(r\in \pi (p!)\).

     

Proof

By Lemma 11, the equation \(q^x\equiv 1\)(mod p) has solutions. Suppose the order of q modulo p is written by h(q). If \(h(q)=p-1\), then q is a primitive root of modulo p. By Lemma 11, we have \(h(q)\mid p-1\). By Lemma 10, we can get s(q). If \(h(q)>s(q)\), then \(q^{h(q)}\mid |L|\), a contradiction to the hypotheses. Then we can assume that \(h(q)\le s(q)\). We can get the q and h(q) by GAP (2016) as Table 1 (Note that there is certain prime which has order \(h(q)<p-1\), but \(h(q)>s(q)\). Hence we do not list in this table).

By NC Theorem, the factor group \(\frac{N_L(P)}{C_L(P)}\) is isomorphic to a subgroup of \(\mathrm {Aut}(P)\cong \mathbb {Z}_{p-1}\) where \(\mathbb {Z}_n\) is a cyclic group of order n. It follows that the order of \(\frac{N_L(P)}{C_L(P)}\) is less than or equal to \(p-1\). If \(11\le q<p\) and \(q^{s(q)}\mid |N_L(P)|\) where \(\exp (|L|,q)=s(q)\), then \(q\mid |C_L(P)|\). This forces \(q\sim p\), a contradiction. This ends the proof of (1).

Next, assume that \(p\in \{ 113, 139, 199, 211, 241, 283, 293, 337, 467, 509, 619, 787, 797, 839, 863, 887, 953, 997\}\). If p divides the order of \(N_L(Q)\), then by NC theorem and Table 1, \(p\mid |C_L(Q)|\) and so \(p\sim q\), a contradiction. This proves (2). (3) follows from Table 1.

This completes the proof of Lemma 12. \(\square\)

Table 1

The values of p and h(q)

p

h(q)

Condition

p

h(q)

Condition

113

\(2^4.7\)

None

139

2.3.23

None

181

\(2^2.3^2.5\)

\(q\ne 19\)

181

4

\(q=19\)

199

\(2.3^2.11\)

None

211

2.3.5.7

None

241

\(2^4.3.5\)

None

283

2.3.47

None

293

\(2^2.73\)

None

317

\(2^2.79\)

\(q\ne 73\)

317

4

\(q=73\)

337

\(2^4.3.7\)

None

409

\(2^3.3.17\)

\(q\ne 31,53\)

409

8

\(q=31\)

409

3

\(q=53\)

421

\(2^2.3.5.7\)

\(q\ne 29\)

421

4

\(q=29\)

467

2.233

None

509

\(2^2.127\)

none

523

\(2.3^2.29\)

\(q\ne 11,19,61\)

523

29

\(q=11\)

523

9

\(q=19\)

523

6

\(q=61\)

547

2.3.7.13

\(q\ne 11,13,41\)

547

39

\(q=11\)

547

21

\(q=13\)

547

6

\(q=41\)

577

\(2^6.3^2\)

\(q\ne 23\)

577

8

\(q=23\)

619

2.3.103

None

631

\(2.3^2.5.7\)

\(q\ne 43\)

631

3

\(q=43\)

661

\(2^3.3.5.11\)

\(q\ne 11\)

661

33

\(q=11\)

691

2.3.5.23

\(q\ne 89\)

691

5

\(q=89\)

709

\(2^2.3.59\)

\(q\ne 227\)

709

3

\(q=227\)

787

2.3.131

None

797

\(2^2.199\)

None

811

\(2.3^4.5\)

\(q\ne 131\)

811

6

\(q=131\)

829

\(2^2.3^2.23\)

\(q\ne 11\)

829

23

\(q=11\)

839

2.419

None

863

2.431

None

887

2.443

None

919

\(2.3^3.17\)

\(q\ne 53\)

919

6

\(q=53\)

953

\(2^3.7.17\)

None

997

\(2^2.3.83\)

None

   

Proof of the main theorem

In this section, we first give the proof of Theorem 4 and second prove Theorem 5.

The proof of Theorem 4

Proof

We divides the proof into two steps.

Step 1 Let \(M=A_{189}\). Assume that G is a finite group such that
$$|G|=|M|$$
and
$$D(G)=D(M).$$
By Lemma 7, the degree pattern GK(G) of G is connected, in particular, the degree pattern GK(G) is the same as the degree pattern of GK(M).

Lemma 13

Let K be a maximal normal soluble subgroup of G. Then K is a \(\{2, 3, 5, 7\}\)-group, in particular, G is insoluble.

Proof

Assume the contrary. First we show that K is a \(181'\)-group. We assume that K contains an element x of order 181. Let C be the centralizer of x in G and N be the normalizer of x in G. It is easy to see from D(G) that C is a \(\{2, 3, 5, 7, 181\}\)-group. By NC theorem, N/C is isomorphic to a subgroup of automorphism group \(\mathrm {Aut}(\langle x\rangle)\cong \mathbb {Z}_{2^2}\times \mathbb {Z}_{3^2}\times \mathbb {Z}_5\), where \(\mathbb {Z}_n\) is a cyclic group of order n. Hence, C is a \(\{2, 3, 5, 7, 181\}\)-group. By Frattini’s arguments, \(G=KN_G(\langle x\rangle)\) and so \(\{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181\}\subseteq \pi (K)\). Since K is soluble, G has a Hall subgroup H of order \(109\cdot 181\). Obviously, \(109\nmid 181-1, H\) is cyclic and so \(109\cdot 181\in \omega (G)\) contradicting \(D(G)=D(M)\).

Second, show that K is a \(p'\)-group, where \(p\in \{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179\}\). Let p be a prime divisor of |K| and P a Sylow p-subgroup of K. By Frattini’s arguments, \(G=KN_G(P)\). It follows from Lemma 12, that 181 is a divisor of \(|N_G(P)|\) if and only if \(p=19\). If \(181\nmid |\mathrm {Aut}(P)|\), then 181 divides the order of \(C_G(P)\) and so there is an element of order \(p\cdot 181\), a contradiction. On the other hand, \(p=19\) and \(181\mid |\mathrm {Aut}(P)|\), where P is the Sylow 19-subgroup of K. By Lemma 10, \(\exp (|L|,19)=9\) and so \(|\frac{N_G(P)}{C_G(P)}|\mid \prod \nolimits _{i=1}^{9}19^{45}\cdot (19^i-1)\). It is easy to get that \(101\nmid \mid \prod \nolimits _{i=1}^{9}19^{45}\cdot (19^i-1)\). If \(101\mid |N_G(P)|\), then 101 is a prime divisor of \(C_G(P)\). Set \(C=C_G(P)\) and \(C_{101}\in \mathrm {Syl}_{101}(C)\). Also \(\exp (|L|,101)=1\). By Frattini’s argument, \(N=CN_N(C_{101})\) and so \(p\nmid |N_N(C_{101})|\). Thus \(181\mid |C|\) and so \(181\sim p\), a contradiction. So \(101\nmid |N_G(P)|\) and \(101\in \pi (K)\). Let \(K_{101}\in \mathrm {Syl}_{101}(K)\). Since \(G=KN_G(K_{101})\), 101 divides the order of \(N_G(K_{101})\), then \(101\nmid |K|\), a contradiction. Therefore K is a \(\{2,3,5,7\}\)-group.

Obviously, \(G\ne K\) and so G is insoluble. \(\square\)

Lemma 14

The quotient group G / K is an almost simple group. More precisely, there is a normal series such that \(S\le G/K\le \mathrm {Aut}(S)\), where S is isomorphic to \(A_{n}\) for \(n\in \{181, 182, 183, 184, 185, 186, 187, 188, 189\}\).

Proof

Let \(H=G/K\) and \(S=\mathrm {Soc}(H)\). Then \(S=B_1\times \cdots \times B_n\), where \(B_i\)’s are non-abelian simple groups and \(S\le H\le \mathrm {Aut}(S)\). In what follows, we will prove that \(n=1\) and \(S\cong A_{n}\).

Suppose the contrary. Obviously, 181 does not divide the order of S, otherwise, there is an element of order \(109\cdot 181\) contradicting \(D(G)=D(A_{189})\). Hence, for every i, we have that \(B_i\in \mathfrak {F}_{179}\), where \(\mathfrak {F}_p\) is the set of non-abelian simple group S with that \(p\in \pi (S)\subseteq \{2,3,\cdots ,p\}\) and p is a prime. But by Lemma 13, K is a \(\{2,3,5,7\}\)-group. Therefore \(181\in \pi (H)\subseteq \pi (\mathrm {Aut}(S))\) and so 181 divides the order of \(\mathrm {Out}(S)\). By Lemma 6, \(\mathrm {Out}(S)=\mathrm {Out}(P_1)\times \cdots \times \mathrm {Out}(P_r)\), where the group \(P_i\)’s are satisfying \(S\cong P_1\times \cdots \times P_r\). Therefore for some j 181 divides the order of an outer-automorphism group of a direct \(P_j\) of t isomorphic simple group \(B_i\). Since \(B_i\in \mathfrak {F}_{179}\), the order of \(\mathrm {Out}(B_i)\) is not divisible by 181 by Lemma 9. By Lemma 6, \(|\mathrm {Aut}(P_j)|=|\mathrm {Aut}(P_j)|^t\cdot t!\). It means \(t\ge 181\), and hence \(4^{181}\mid |G|\), a contradiction. Thus \(n=1\) and \(S=B_1\).

By Lemma 13, we can assume that \(|S|=2^a\cdot 3^b\cdot 5^c\cdot 7^{d}\cdot 11^{18}\cdot 13^{15}\cdot 17^{11}\cdot 19^9\cdot 23^8\cdot 29^6\cdot 31^6\cdot 37^5\cdot 41^4\cdot 43^4\cdot 47^4\cdot 53^4\cdot 59^3\cdot 61^3\cdot 67^2\cdot 71^2\cdot 73^2\cdot 79^2\cdot 83^2\cdot 89^2\cdot 97\cdot 101\cdot 107\cdot 109\cdot 113\cdot 127\cdot 131\cdot 137\cdot 139\cdot 149\cdot 151\cdot 157\cdot 163\cdot 167\cdot 173\cdot 179\cdot 181\), where \(2\le a \le 182, 1\le b\le 93, 1\le c\le 45\) and \(1\le d\le 30\). By Zavarnitsine (2009), the only possible group is isomorphic to \(A_n\) with \(n\in \{181, 182, \ldots , 189\}\).

This completes the proof. \(\square\)

We continue the proof of Theorem 4. By Lemma 14, S is isomorphic to \(A_{n}\) with \(n\in \{181, 182, \cdots , 189\}\), and \(S\le G/K\le \mathrm {Aut}(S)\).

Case 1

Let \(S\cong A_{181}\).

Then \(A_{181}\le G/K\le S_{181}\). If \(G/K\cong A_{181}\), then \(|K|=182\cdot 183\cdot 184\cdot 185\cdot 186\cdot 187\cdot 188\cdot 189=2^6\cdot 3^5\cdot 5\cdot 7^2\cdot 11\cdot 13\cdot 17\cdot 23\cdot 31\cdot 37\cdot 47\) and so \(11,13,17,23,31,37,47\in \pi (K)\) contradicting to Lemma 13.

If \(G/K\cong S_{181}\), we also have that 11, 13, 17 or 19 divides |K|, contradicting to Lemma 13.

Similarly we can rule out these cases “\(S\cong A_{n}\) with \(n\in \{182, 183, \cdots , 187\}\)”.

Case 2

Let \(S\cong A_{188}\).

Then \(A_{188}\le G/K\le S_{188}\). Therefore \(G/K\cong A_{188}\) or \(G/K\cong S_{188}\).
  1. (1.1)

    Let \(G/K\cong A_{188}\). Then \(|K|=7\cdot 3^3\). By Conway et al. (1985), the order of \(\mathrm {Out}(A_{188})\) is 2 and the Schur multiplier of \(A_{188}\) is 2. Then G is isomorphic to \(K\times A_{188}\). By Lemma 8, there are 13 types of groups of order 189 satisfying that \(|G|=|M|\) and \(D(G)=D(M)\).

     
  2. (1.2)

    Let \(G/K\cong S_{188}\). Since \(|S_{188}|_2=|S_{189}|_2>|A_{189}|_2\), then we rule out this case.

     

Case 3

Let \(S\cong A_{189}\).

Then \(A_{189}\le G/K\le S_{189}\). If \(G/K\cong A_{189}\), then order consideration implies that G is isomorphic to \(A_{189}\). If \(G/K\cong S_{189}\), then as \(|S_{189}|_2>|A_{189}|_2=|G|_2\), we rule out this case.

Step 2 Similarly as the proof of (1), the following results are given:
  1. (1)

    K is a maximal soluble normal \(\{2,3,5,7\}\)-group.

     
  2. (2)

    \(S\le G/K\le \mathrm {Aut}(S)\), where S is isomorphic to one of the groups: \(A_{139}, A_{140}, \ldots , A_{146}\) and \(A_{147}\).

     

Case 1

Let \(S\cong A_{139}\).

Then \(A_{139}\le G/K\le S_{139}\). If the former, then \(11\mid |K|\), a contradiction. If the latter, we also have that \(11\mid |K|\) and so we rule out.

Similarly we can rule out these cases “S is isomorphic to \(A_{140}, A_{141}, \ldots , A_{145}\)”.

Case 2

Let \(S\cong A_{146}\).

Then \(A_{146}\le G/K\le S_{146}\). If \(G/K\cong A_{146}\), then \(|K|=3\cdot 7^2\). Since the order of \(\mathrm {Out}(A_{147})\) is 2 and the Schur multiplier of \(A_{147}\) is 2. Then G is isomorphic to \(K\times A_{146}\). By GAP (2016), there are six types of groups of order 147. So there are 6 groups with the hypotheses: \(|G|=|A_{147}|\) and \(D(G)=D(A_{147})\). If \(G/K\cong S_{147}\), then as \(|S_{146}|_2>|A_{146}|_2=|A_{147}|_2=|G|_2\), we rule out.

Case 3

Let \(S\cong A_{147}\).

Then \(A_{147}\le G/K\le S_{147}\). If the former, then \(K=1\) and so \(G\cong A_{147}\), the desired result. If the latter, then as \(|S_{147}|_2>|A_{147}|_2=|G|_2\), we rule out.

We also can get that \(A_{147}\) is 7-fold OD-characterizable.

This completes the proof of Theorem 4. \(\square\)

The proof of Theorem 5

Proof

Assume that \(|G|=|A_{p+8}|\) and \(D(G)=D(A_{p+8})\), then by Lemma 7, the degree pattern GK(G) of G is the same as \(GK(A_{p+8})\) of \(A_{p+8}\). Similarly as the proof of Theorem 4, the statements are gotten:
  1. (1)

    Let K be a maximal soluble group. Then K is a \(\{2, 3, 5, 7\}\)-group, in particular, G is insoluble.

     
  2. (2)

    There is a normal series such that \(S\le G/K\le \mathrm {Aut}(S)\), where S is isomorphic to \(A_{p+r}\) with that \(0\le r\le 8\) and \(p\in \{\)113, 139, 199, 211, 241, 283, 293, 317, 337, 409, 421, 467, 509, 523, 547, 577, 619, 631, 661, 691, 709, 787, 797, 811, 829, 839, 863, 887, 919, 953, 997\(\}\).

     
In what follows, we consider the case “\(p=113\)”.
  1. (1)

    \(S\cong A_{113}\).

    Then \(A_{113}\le G/K\le S_{113}\). If \(G/K\cong A_{113}\), then 11 divides the order of K, a contradiction. If \(G/K\cong S_{113}\), then we also have that \(11\mid |K|\), a contradiction. Similarly we can get a contradiction when S is isomorphic to one of \(A_{114}, A_{115}, A_{116}, A_{117}, A_{118}, A_{119}\), and \(A_{120}\).

     
  2. (2)

    Let \(S\cong A_{121}\).

    Then \(A_{121}\le G/K\cong S_{121}\). If \(G/K\cong A_{121}\), then \(K=1\), the desired result. If \(G/K\cong S_{121}\), then as \(|S_{121}|_2>|G|_2=|A_{121}|_2\), a contradiction.

     
Similarly we can deal with these cases “\(p\in \{139, 199, 211, 241, 283, 293, 317, 337, 409, 421, 467, 509, 523, 547, 577, 619, 631, 661, 691, 709, 787, 797, 811, 829, 839, 863, 887, 919, 953, 997\}\)”.

This completes the proof of Theorem 5. \(\square\)

Non OD-characterization of some alternating groups

Assume that p is a prime and m is an integer larger than 3. If \(\pi ((p+m)!)\subseteq \pi (p!)\), then \(GK(A_{p+m})\) is connected.

For the alternating group \(A_{p+m}, |A_{p+m}|=(p+m)|A_{p+m-1}|\).

We shall use the notation v(n) to denote the number of types of groups of order n where n is a positive integer. We follows the method of Moghaddamfar (2015), \(h_{OD}(A_{p+m})\ge 1+v(p+m)\) where \(\pi (A_{p+m})=\pi (A_p)\) and \(m\ge 1\) is a non-prime integer. We get the results as Table 2 which contains some results of Liu and Zhang (Submitted), Moghaddamfar (2015), Mahmoufifar and Khosravi (2014).
Table 2

Non OD-characterizability of alternating groups

G

p

m!

\(\pi (p+m)\)

\(h_{OD}\)

References

\(A_{125}\)

113

12!

\(\{5\}\)

≥6

Mahmoufifar and Khosravi (2014)

\(A_{147}\)

139

8!

\(\{3,7\}\)

≥7

Moghaddamfar (2015)

\(A_{189}\)

181

8!

\(\{3,7\}\)

≥14

Moghaddamfar (2015)

\(A_{539}\)

523

16!

\(\{7,11\}\)

≥3

Moghaddamfar (2015)

\(A_{625}\)

619

6!

\(\{5\}\)

≥16

Moghaddamfar (2015)

\(A_{875}\)

863

12!

\(\{13,67\}\)

≥6

Moghaddamfar (2015)

\(A_{1029}\)

1019

10!

\(\{3,7\}\)

≥20

 

\(A_{1144}\)

1129

15!

\(\{2,11,13\}\)

≥40

 

\(A_{1274}\)

1159

15!

\(\{2,7,13\}\)

≥11

 

\(A_{1344}\)

1319

25!

\(\{2,3,7\}\)

≥11,721

 

\(A_{1352}\)

1319

33!

\(\{2,13\}\)

≥53

 

Note that v(n), the number of groups of given small order n can be computed by GAP (2016). The Gap programme is as followings.

gap> SmallGroupsInformation(n);

So we have the following conjecture.

Conjecture Assume that p is a prime and \(m\ge 6\) is not a prime. If \(\pi ((p+m)!)\subseteq \pi (p!)\) and \(\pi (p+m)\subseteq \pi (m!)\), then \(A_{p+m}\) is not OD-characterizable.

Conclusion

In this paper, we have proved the following two results.

Result 1a: The alternating group \(A_{189}\) of degree 189 is 14-fold OD-characterizable.

Result 1b: The alternating group \(A_{147}\) of degree 147 is 7-fold OD-characterizable.

Result 2: Let p be a prime with the following three conditions:
  1. (1)

    \(p\ne 139\) and \(p\ne 181\),

     
  2. (2)

    \(\pi ((p+8)!)=\pi (p!)\),

     
  3. (3)

    \(p\le 997\).

     
Then the alternating group \(A_{p+8}\) of degree \(p+8\) is OD-characterizable.

Notes

Declarations

Authors' contributions

SL and ZZ contributed this paper equally. Both authors read and approved the final manuscript.

Acknowlegements

The first author was supported by the Opening Project of Sichuan Province University Key Laborstory of Bridge Non-destruction Detecting and Engineering Computing (Grant Nos: 2013QYJ02 and 2014QYJ04); the Scientific Research Project of Sichuan University of Science and Engineering (Grant No: 2014RC02) and by the department of Sichuan Province Eduction(Grant Nos: 15ZA0235 and 16ZA0256). The authors are very grateful for the helpful suggestions of the referee.

Dedicated to Prof Gui Min Wei on the occasion of his 70th birthday.

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Science, Sichuan University of Science and Engineering
(2)
Sichuan Water Conservancy Vocational College

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