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CUBO A Mathematical Journal
Vol.13, No¯ 02, (151–161). June 2011

Some New Characterizations for PGL(2, q)

B. Khosravi 12, M. Khatami2 and Z. Akhlaghi2

1 School of Mathematics,

Institute for Research in Fundamental Sciences (IPM),

P.O. Box: 19395-5746, Tehran, Iran.

email: khosravibbb@yahoo.com

and
2 Dept. of Pure Math., Faculty of Math. and Computer Sci.,

Amirkabir University of Technology (Tehran Polytechnic),

424, Hafez Ave., Tehran 15914, IRAN.

ABSTRACT

Many authors introduced some characterizations for finite groups. In this paper as

the main result we prove that the finite group PGL(2, q) is uniquely determined by its

noncommuting graph. Also we prove that PGL(2, q) is characterizable by its noncyclic

graph. Throughout the proof of these results we prove that PGL(2, q) is uniquely

determined by its order components and using this fact we give positive answer to a

conjecture of Thompson and another conjecture of Shi and Bi for the group PGL(2, q).

RESUMEN

Muchos autores introdujeron algunas caracterizaciones de los grupos finitos. En este

trabajo como principal resultado se demuestra que grupo finito PGL(2, q) es determi-

nado nicamente por su gráfica no conmutativa. También se demuestra que PGL(2, q)

1The First author was supported in part by a grant from IPM (no. 89200113).



152 B. Khosravi, M. Khatami & Z. Akhlaghi CUBO
13, 2 (2011)

es caracterizable por su gráfico no ćıclico. A lo largo de la prueba de estos resultados se

demuestra que PGL(2, Q) es determinado únicamente por los componentes de su orden

y con ello damos respuesta positiva a una conjetura de Thompson y otra conjetura de

Shi Bi y para el grupo PGL(2, q).

Keywords and phrases: Noncommuting graph, prime graph, noncyclic graph, order components.

Mathematics Subject Classification: 20D05, 20D60.

1 Introduction

If n is an integer, then we denote by π(n) the set of all prime divisors of n. If G is a finite group,

then π(|G|) is denoted by π(G). We construct the prime graph of G which is denoted by Γ (G) as

follows: the vertex set is π(G) and two distinct primes p and q are joined with an edge if and only

if G contains an element of order pq. Let t(G) be the number of connected components of Γ (G)

and let π1, π2, ..., πt(G) be the connected components of Γ (G). If 2 ∈ π(G), then we assume that
2 ∈ π1.

Now we can express |G| as a product of coprime natural numbers mi, such that 1 ≤ i ≤ t(G)
and π(mi) = πi. These integers are called order components of G. The set of order components of

G is denoted by OC(G).

One of the other graphs which associated with a non-abelian group G is the noncommuting

graph that is denoted by ∇(G) and is constructed as follows: the vertex set of ∇(G) is G \ Z(G)
with two vertices x and y are joined by an edge whenever the commutator of x and y is not identity.

In [1] the authors put forward the following conjecture:

Conjecture A. Let S be a finite non-abelian simple group and G be a finite group such that

∇(G) ∼= ∇(S). Then G ∼= S.
The validity of this conjecture has been proved for all simple groups with non-connected prime

graphs. Also it is proved that some finite simple groups with connected prime graphs, say A10,

U4(7), L4(8), L4(4) and L4(9), can be uniquely determined by their noncommuting graghs (see

[19, 20, 21, 22]).

In this paper as the main result we prove that the almost simple group PGL(2, q), where

q = pn for a prime number p and a natural number n, is characterizable by its noncommuting

graph. As a consequence of our results we prove the validity of a conjecture of Thompson and

another conjecture of Shi and Bi for the group PGL(2, q).

Let G be a noncyclic group and Cyc(G) = {x ∈ G|〈x, y〉 is cyclic for all y ∈ G}. In [2], the
authors introduced the cyclic graph of G, which is denoted by Γ1(G) as follows: take G \ Cyc(G)

as the vertex set and join two vertices if they do not generate a cyclic subgroup. In this graph the

degree of each vertex x is equal to |G| \ |CycG(x)|, where CycG(x) = {y ∈ G|〈x, y〉 is cyclic}. It is



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Some New Characterizations for PGL(2, q) 153

proved that some finite simple groups, Sn, D2k , D2n ,where n is odd, are characterizable by the

noncyclic graph. We show that PGL(2, q) is uniquely determined by its noncyclic graph.

In this paper, all groups are finite and by simple groups we mean non-abelian simple groups.

All further unexplained notations are standard and refer to [6], for example.

2. Preliminary results

In this section we bring some preliminary lemmas which are necessary in the proof of the main

theorem.

Remark 2.1. Let N be a normal subgroup of G and p, q be incident vertices of Γ (G/N). Then p, q

are incident in Γ (G). In fact if xN is of order pq, then there exists a power of x which is of order pq.

Definition 2.2. ([8]) A finite group G is called a 2-Frobenius group if it has a normal series

1 � H � K � G, where K and G/H are Frobenius groups with kernels H and K/H, respectively.

Lemma 2.3. Let G be a Frobenius group of even order and let H, K be Frobenius comple-

ment and Frobenius kernel of G, respectively. Then t(G) = 2, and the prime graph components of

G are π(H), π(K) and G has one of the following structures:

(a) 2 ∈ π(K) and all Sylow subgroups of H are cyclic;
(b) 2 ∈ π(H), K is an abelian group, H is a solvable group, the Sylow subgroups of odd order of
H are cyclic groups and the 2-Sylow subgroups of H are cyclic or generalized quaternion groups;

(c) 2 ∈ π(H), K is an abelian group and there exists H0 ≤ H such that |H : H0| ≤ 2, H0 =
Z × SL(2, 5), π(Z) ∩ {2, 3, 5} = ∅ and the Sylow subgroups of Z are cyclic.

Also the next lemma follows from [8] and the properties of Frobenius groups [9]:

Lemma 2.4. Let G be a 2-Frobenius group, i.e., G has a normal series 1 � H � K � G, such that

K and G/H are Frobenius groups with kernels H and K/H, respectively. Then

(a) t(G) = 2, π1 = π(G/K) ∪ π(H) and π2 = π(K/H);
(b) G/K and K/H are cyclic, |G/K| | (|K/H| − 1) and G/K ≤ Aut(K/H);
(c) H is nilpotent and G is a solvable group.

Lemma 2.5. ([4, Lemma 8]) Let G be a finite group with t(G) ≥ 2 and let N be a normal
subgroup of G. If N is a πi-group for some prime graph component of G and m1, m2, . . . , mr are

some order components of G but not πi-numbers, then m1m2 · · · mr is a divisor of |N| − 1.

Lemma 2.6. ([3, Lemma 1.4]) Suppose G and M are two finite groups satisfying t(M) ≥ 2,
N(G) = N(M), where N(G)={n | G has a conjugacy class of size n }, and Z(G) = 1. Then



154 B. Khosravi, M. Khatami & Z. Akhlaghi CUBO
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|G| = |M|.

Lemma 2.7. ([3, Lemma 1.5]) Let G1 and G2 be finite groups satisfying |G1| = |G2| and

N(G1) = N(G2). Then t(G1) = t(G2) and OC(G1) = OC(G2).

Lemma 2.8. ([11]) Let G be a finite group and M be a finite group with t(M) = 2 satisfy-

ing OC(G) = OC(M). Let OC(M) = {m1, m2}. Then one of the following holds:

(a) G is a Frobenius or 2-Frobenius group;

(b) G has a normal series 1 � H � K � G such that G/K is a π1-group, H is a nilpotent π1-group,

and K/H is a non-abelian simple group. Moreover OC(K/H) = {m′1, m
′

2, . . . , m
′

s, m2}, where

m′1m
′

2 . . . m
′

s|m1. Also G/K ≤ Out(K/H).

Lemma 2.9. ([1]) Let G be a finite non-abelian group. If H is a group such that ∇(G) ∼= ∇(H),
then H is a finite non-abelian group such that |Z(H)| divides

gcd(|G| − |Z(G)|, |G| − |CG(x)|, |CG(x)| − |Z(G)| : x ∈ G \ Z(G)).

Lemma 2.10. ([18]) Let G be a non-abelian group such that ∇(G) ∼= ∇(PSL(2, 2n)), where n is
a natural number. Then G ∼= PSL(2, 2n).

Lemma 2.11.([7, Remark 1]) The equation pm − qn = 1, where p and q are primes and m, n > 1

has only one solution, namely 32 − 23 = 1.

Lemma 2.12. ([2]) Let G be a finite noncyclic group. If H is a group such that Γ1(G) ∼= Γ1(H),

then H is a finite noncyclic group such that |Cyc(H)| divides

gcd(|G| − |Cyc(G)|, |G| − |CycG(x)|, |CycG(x)| − |Cyc(G)| : x ∈ G \ Cyc(G)).

Lemma 2.13. ([2]) Let G and H be two finite noncyclic groups such that Γ1(G) ∼= Γ1(H). If

|G| = |H|, then πe(G) = πe(H).

3. Main Results

We note that if q = 2n, then PGL(2, q) = PSL(2, q) and we know that PSL(2, q) is characterizable

by its noncommuting graph (see [18]). Therefore throughout this section we suppose M is the

almost simple group PGL(2, q), where q = pn for an odd prime number p and a natural number

n.

Theorem 3.1. Let G be a group such that ∇(G) ∼= ∇(M). Then |G| = |M|.

Proof. First note that G is a finite non-abelian group. Since ∇(G) ∼= ∇(M), we have |G|−|Z(G)| =
|M| − |Z(M)|. Then it is enough to prove that |Z(G)| = |Z(M)|.



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Some New Characterizations for PGL(2, q) 155

By Lemma 2.9, |Z(G)| divides |M| − |Z(M)|. Since |Z(M)| = 1, we have |Z(G)| divides q(q2 −

1) − 1. Let P be a Sylow p-subgroup of M. We know that Z(P) 6= 1. So there exists 1 6= x ∈ Z(P).
We claim that CM(x) = P. It is obvious that P ≤ CM(x), since x ∈ Z(P). On the contrary

we suppose that y ∈ CM(x) \ P. So we can conclude that o(xy) = o(x)o(y). Without lose of
generality we suppose |y| = r, where r 6= p is a prime number. Then M has an element of order
rp. But p is an isolated vertex in Γ (M), a contradiction. Therefore our claim is proved.

By Lemma 2.9 we have |Z(G)| divides |CM(x)| − |Z(M)|. Then |Z(G)| divides q − 1. We know

that Z(G) divides q(q2 − 1) − 1, which implies that |Z(G)| = 1 and so |G| = |M|. 2

Theorem 3.2. Let G be a group such that ∇(G) ∼= ∇(M), where M = PGL(2, q). Then
OC(G) = OC(M).

Proof. Since ∇(G) ∼= ∇(M), the set of vertex degrees of two graphs are the same. Therefore

{|G| − |CG(x)| | x ∈ G} = {|M| − |CM(y)| | y ∈ M}.

On the other hand Theorem 3.1 implies that |G| = |M|, and so N(G) = N(M). Now using Lemma

2.7 we have OC(G) = OC(M). 2

Theorem 3.3. Let G be a finite group and OC(G) = OC(M). If q = pn 6= 3 then G is nei-
ther a Frobenius group nor a 2-Frobenius group. If q = 3 and G is a 2-Frobenius group, then

G ∼= S4.

Proof. If G is a Frobenius group, then by Lemma 2.3, OC(G) = {|H|, |K|} where K and H are

Frobenius kernel and Frobenius complement of G, respectively. Therefore OC(G) = {q, q2 − 1} and

since |H| | (|K| − 1) it follows that |H| < |K| and so |H| = q and |K| = q2 − 1. Also q | (q2 − 2)

implies that q = 2, which is a contradiction, since q is odd. Therefore G is not a Frobenius group.

Let G be a 2-Frobenius group. Hence G = ABC, where A and AB are normal subgroups of G;

AB and BC are Frobenius groups with kernels A, B and complements B, C, respectively. By Lemma

2.4, we have |B| = q and |A||C| = q2 − 1. Also |B| | (|A| − 1) and so |A| = qt + 1, for some t > 0.

On the other hand, |A| | (q2 − 1), which implies that q2 − 1 = k(qt + 1), for some k > 0. Therefore

q | (k + 1) and so q − 1 ≤ k. If t > 1, then q2 − 1 = k(qt + 1) ≥ (q − 1)(qt + 1) > (q − 1)(q + 1),
which is a contradiction. Hence t = 1 and |A| = q + 1 and |C| = q − 1.

If there exists an odd prime r such that r | (q + 1), then let R be a Sylow r-subgroup of

A. Since A is a nilpotent group, it follows that R is a normal subgroup of G. Now Lemma 2.5,

implies that q | (|R| − 1) and |R| | (q + 1)/2, which is a contradiction. Therefore q + 1 = 2α, for

some α > 0. Similarly Z(A) 6= 1 is a characteristic subgroup of A and hence A is abelian. Let
X = {x ∈ A|o(x) = 2} ∪ {1}. Then X is a non-identity characteristic subgroup of A. Therefore A
is an elementary abelian 2-subgroup of G and |A| = 2α = q + 1. By Lemma 2.11, if q = pn such

that n > 1, then the equation 2α − q = 1 does not have any solution.



156 B. Khosravi, M. Khatami & Z. Akhlaghi CUBO
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Now let n = 1. Suppose F = GF(2α) and so A is the additive group of F. Also |B| = q =

p = 2α − 1 and so B is the multiplicative group of F. Now C acts by conjugation on A and

similarly C acts by conjugation on B and this action is faithful. Therefore C keeps the struc-

ture of the field F and so C is isomorphic to a subgroup of the automorphism group of F. Hence

|C| = 2α − 2 ≤ |Aut(F)| = α. Therefore α ≤ 2. If α = 2, then G = S4, the symmetric group on 4
letters. 2

Lemma 3.4. Let G be a finite group and M = PGL(2, q), where q > 3 or q = 3 and M is

not a 2-Frobenius group. If OC(G) = OC(M), then G has a normal series 1 � H � K � G such

that H and G/K are π1-groups and K/H is a simple group. Moreover the odd order component

of M is equal to an odd order component of K/H. In particular, t(K/H) ≥ 2. Also |G/H| divides
|Aut(K/H)|, and in fact G/H ≤ Aut(K/H).

Proof. The first part of the lemma follows from Lemma 2.8 and Theorem 3.3, since the prime

graph of G has two components. If K/H has an element of order pq, where p and q are primes,

then by Remark 2.1, K has an element of order pq. Therefore G has an element of order pq. So by

the definition of prime graph component, the odd order component of G is equal to an odd order

component of K/H. Also K/H � G/H and CG/H(K/H) = 1, which implies that

G/H =
NG/H(K/H)

CG/H(K/H)
∼= T , T ≤ Aut(K/H). 2

Theorem 3.5. Let G be a finite group such that OC(G) = OC(M), where M = PGL(2, q).

Then G ∼= PGL(2, q).

Proof. If q = 3 and G is a 2-Frobenius group, then Theorem 3.3 implies that G = S4 ∼= PGL(2, 3),

as desired. Otherwise Lemma 3.4 implies that G has a normal series 1 � H � K � G such that H

and G/K are π1-groups and K/H is a simple subgroup and t(K/H) ≥ 2.
Now using the classification of finite simple groups and the results in Tables 1-3 in [10], we

consider the following cases.

Case 1. Let K/H ∼= Am, where m = p
′, p′ + 1 or p′ + 2 and p′ ≥ 5 is a prime number and

m and m − 2 are not primes at the same time.

Then q = p′, and consequently n = 1 and q = p = p′. On the other hand, |Am| | |G| =

p(p2 − 1). If m > p, then |Am| > (p + 1)p(p − 1), which is a contradiction. Therefore m = p and

|Ap| | |G| = p(p
2 − 1), and so |Ap| = p!/2 ≤ p(p2 − 1). Hence (p − 2)!/2 ≤ p + 1. But p ≥ 7,

since p − 2 is not a prime. So (p − 2)(p − 3) < (p − 2)!/2 ≤ p + 1, which is a contradiction. This
completes the proof.

Case 2. Let K/H ∼= Ap ′ , where p
′ and p′ − 2 are primes.



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Some New Characterizations for PGL(2, q) 157

If p = p′, for p′ ≥ 7, then we can get a contradiction similarly to the previous case. So
p = 5 and K/H ∼= A5 ∼= PSL(2, 5). Since K/H ≤ G/H ≤ Aut(K/H), we have PSL(2, 5) ≤ G/H ≤
PGL(2, 5). Hence G/H is isomorphic to PSL(2, 5) or PGL(2, 5). If G/H ∼= PSL(2, 5), then |H| = 2.

But H � G, which implies that H ⊆ Z(G) and we get a contradiction. So G/H ∼= PGL(2, 5), which
implies that H = 1 and G ∼= PGL(2, 5).

Let p = p′ − 2. Since p′ | |Ap ′ |, we have p
′ | |G| = p(p2 − 1). But we know that p = p′ − 2 is

the greatest prime divisor of |G|, which is a contradiction.

Case 3. Let K/H be a sporadic simple group.

Using the tables in [10] we see that the odd order components of sporadic simple groups are

prime.

Let S be a sporadic simple group and K/H ∼= S. Since q is equal to the greatest odd order

component of K/H, we have q = mi, such that mi = max{m2, m3, ..., mt(S)}. So q is a prime

number.

If S = J4, then q = p = 43. Since 11
2 | |K/H|, we have 112 | (p2 − 1) = 432 − 1, which is a

contradiction.

If S = Co2, then q = p = 23. Since 7 | |K/H|, we have 7 | (23
2 − 1), which is a contradiction.

The proof of other cases are similar and we omit them for convenience.

If K/H is isomorphic to 2A3(2),
2F4(2)

′, A2(4),
2A5(2), E7(2), E7(3) or

2E6(2), then similarly

we get a contradiction.

In the sequel of the proof we consider simple groups of Lie type. Since the proofs of these

cases are similar we state only a few of them.

In all of the following cases p′ is an odd prime number and q′ is a prime power.

Case 4. Let K/H ∼= Ap ′−1(q
′), where (p′, q′) 6= (3, 2), (3, 4). By hypothesis we have q =

(q′p
′

− 1)/((q′ − 1)(p′, q′ − 1)). Hence q < q′p
′

− 1 < q′p
′

. Then q2 − 1 < q′2p
′

. On the other

hand, we know q′p
′
(p

′
−1)/2 | (q2 − 1) and therefore q′p

′
(p

′
−1)/2 < q′2p

′

. So p′(p′ − 1)/2 < 2p′

and hence p′ < 5. So p′ = 3 and q = (q′2 + q′ + 1)/(3, q′ − 1), which implies that q < 2q′2.

Therefore q2 − 1 < 4q′4 − 1. On the other hand, q′3(q′2 − 1)(q′ − 1) | (q2 − 1) and consequently

q′3(q′2 − 1)(q′ − 1) < 4q′4 − 1. So q′ = 2, 3 or 4. Since (p′, q′) 6= (3, 2), (3, 4), we have q′ = 3
and q = 13. Then 33(32 − 1)(3 − 1) | (132 − 1), which is a contradiction.

Case 5. Let K/H ∼= 2Ap ′ (q
′), where (q′ + 1) | (p′ + 1) and (p′, q′) 6= (3, 3), (5, 2). In this

case we have q = (q′p
′

+ 1)/(q′ + 1). Therefore q < q′p
′

+ 1 < 2q′p
′ ≤ q′p ′+1 and hence

q2 − 1 < q′2(p
′
+1). On the other hand, we have q′p

′
(p

′
+1)/2 | (q2 − 1). So we conclude that

q′p
′
(p

′
+1)/2 < q′2(p

′
+1). Hence p′(p′ + 1)/2 < 2(p′ + 1), which implies that p′ = 3. Then

(q′ + 1) | 4 and hence q′ = 3. So (p′, q′) = (3, 3), which is impossible.



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Case 6. Let K/H ∼= Bn(q
′), where n = 2m ≥ 4 and q′ is odd. Therefore q = (q′n + 1)/2.

So q < 2q′n < q′n+1. Therefore q2 − 1 < q′2(n+1). On the other hand, we have q′n
2

| (q2 − 1)

and consequently q′n
2

< q′2(n+1). So n2 < 2(n + 1), which implies that n = 2, and this is a

contradiction.

Case 7. Let K/H ∼= Cn(q
′), where n = 2m ≥ 2. Then q = (q′n + 1)/(2, q′ − 1). Therefore

q ≤ q′n + 1 < 2q′n ≤ q′n+1, which implies that q2 − 1 < q′2(n+1). On the other hand, we have
q′n

2

| (q2 − 1), which implies that q′n
2

< q′2(n+1). So we have n2 < 2(n + 1) and hence n = 2.

Therefore q < 2q′2 and so q′4(q′2 − 1) < q2 − 1 < 4q′4, which is impossible.

Case 8. Let K/H ∼= 2Dp ′ (3), where p
′ = 2n + 1 ≥ 5. So we have q = (3p ′ + 1)/4 or

q = (3p
′
−1 + 1)/2.

If q = (3p
′

+ 1)/4, then q < 3p
′
+1. On the other hand, we have 3p

′
(p

′
−1) | (q2 − 1), which

implies that 3p
′
(p

′
−1) ≤ q2 − 1 < 32(p ′+1). Therefore p′(p′ − 1) < 2(p′ + 1), and hence p′ ≤ 3,

which is impossible.

If q = (3p
′
−1 + 1)/2, then q < 3p

′

. On the other hand, 3p
′
(p

′
−1) | (q2 − 1), which implies

that 3p
′
(p

′
−1) < 32p

′

, and so p′(p′ − 1) < 2p′, which is impossible.

Case 9. Let K/H ∼= 2B2(q
′), where q′ = 22n+1 > 2. In this case we have q = q′ ±

√
2q′ + 1 or

q = q′ − 1.

If q = q′ ±
√

2q′ + 1, then q2 − 1 = q′2 + 4q′ ± 2
√

2q′(q′ + 1). On the other hand, we have

q′2 | (q2 − 1) and so q′ | (q′2 + 4q′ ± 2
√

2q′(q′ + 1)), which implies that q′ ≤ 2
√

2q′. Hence

q′2 ≤ 8q′. Therefore q′ = 8 and so q = 5 or 13, which is a contradiction by q′2 | (q2 − 1).
If q = q′ − 1, then q′2|(q′2 − 2q′), which is a contradiction.

Case 10. Let K/H ∼= 2F4(q
′), where q′ = 22n+1 > 2. In this case we have q = q′2 ±

√

2q′3 +

q′ ±
√

2q′ + 1. Therefore q < 4q′2 < q′3 and so q2 − 1 < q′6. On the other hand, q′12 | (q2 − 1),

which is a contradiction.

Case 11. Let K/H ∼= A1(q
′), where 4|q′. By hypothesis we have q = q′ − 1 or q = q′ + 1.

If q = q′ − 1, then q2 − 1 = q′2 − 2q′. But we know q′(q′ + 1) | (q2 − 1), which is a

contradiction.

If q = q′ + 1, then q2 − 1 = q′2 + 2q′. Since q′(q′ − 1) | (q2 − 1), we conclude that (q′ − 1) | 3.

So q′ = 4 and hence K/H ∼= A1(4) ∼= A5. By the proof of Case 2 we have K/H ∼= PGL(2, 5).

Case 12. If K/H ∼= A1(q
′), where 4|(q′ − 1), then q = (q′ + 1)/2 or q = q′.

If q = (q′ + 1)/2, then q2 − 1 = (q′2 − 3 + 2q′)/4. On the other hand, q′(q′ − 1) | (q2 − 1)



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Some New Characterizations for PGL(2, q) 159

and hence q′(q′ − 1) ≤ (q′2 − 3 + 2q′)/4. So q′2 − 2q′ + 1 ≤ 0, which is a contradiction.
If q = q′, then K/H ∼= A1(q) = PSL(2, q). Since K/H ≤ G/H and |G| = 2|PSL(2, q)|, we

conclude that |H| = 1 or 2. Let |H| = 2. Since H � G we have H ⊆ Z(G), which is a contradiction.
So H = 1.

By Lemma 2.8, G/K ≤ Out(K/H) and |G/K| = 2. If G/K contains a field automorphism, then
2p ∈ πe(G), which is a contradiction. If G/K contains a diagonal-field automorphism, then G is
the non-split extension of PSL(2, q) by Z2 and we know that the prime graph of G is the prime

graph of PSL(2, q) (see [12]), which is a contradiction. So a diagonal automorphism generates G/K

and consequently G ∼= PGL(2, q).

If K/H ∼= A1(q
′), where 4|(q′ + 1), then similarly we conclude that G ∼= PGL(2, q). 2

Theorem 3.6. Let G be a group such that ∇(G) ∼= ∇(M), where M = PGL(2, q) and q is a
prime power. Then G ∼= M.

Proof. If q = 2n, where n is an integer, then PGL(2, q) ∼= PSL(2, q) and so Lemma 2.10 im-

plies that G ∼= M. If q is odd, then obviously the theorem follows from Theorems 3.2 and 3.5. 2

Remark 3.7. It is a well known conjecture of J. G. Thompson that if G is a finite group with

Z(G) = 1 and M is a non-abelian simple group satisfying N(G) = N(M), then G ∼= M.

We can give a positive answer to this conjecture for the group PGL(2, q) by our characteriza-

tion of this group.

Corollary 3.8. Let G be a finite group with Z(G) = 1 and M = PGL(2, q), where q is a

prime power. If N(G) = N(M), then G ∼= M.

Proof. By Lemmas 2.6 and 2.7, if G and M are two finite groups satisfying the conditions of

Corollary 3.8, then OC(G) = OC(M). So using Theorem 3.5 we get the result. 2

Remark 3.9. W. Shi and J. Bi in [16] put forward the following conjecture:

Conjecture. Let G be a group and M be a finite simple group. Then G ∼= M if and only if

(i) |G| = |M|, and,

(ii) πe(G) = πe(M), where πe(G) denotes the set of orders of elements in G.

This conjecture is valid for sporadic simple groups [13], alternating groups [17], and some

simple groups of Lie type [14, 15, 16]. As a consequence of Theorem 3.5, we prove the validity of

this conjecture for the almost simple group PGL(2, q), where q is a prime power.

Corollary 3.10. Let G be a finite group and M = PGL(2, q), where q is a prime power. If



160 B. Khosravi, M. Khatami & Z. Akhlaghi CUBO
13, 2 (2011)

|G| = |M| and πe(G) = πe(M), then G ∼= M.

Proof. By assumption we have OC(G) = OC(M). Thus the corollary follows from Theorem

3.5. 2

Proposition 3.11. Let G be a group such that Γ1(G) ∼= Γ1(M), where M = PGL(2, q) and

q is a prime power. Then G ∼= M.

proof. First we show that |G| = |M|. By Lemma 2.12 we have |Cyc(G)| divides |M| − |Cyc(M)|.

Since Cyc(M) ≤ Z(M) = 1, it follows that |Cyc(G)| divides |M|−1. On the other hand, by Lemma
2.12, |Cyc(G)| divides |CycM(x)| − |Cyc(M)|, where x ∈ M \ Cyc(M). Let x be a p-element of
M. We claim that 〈x〉 = CycM(x). We know that 〈x〉 ⊆ CycM(x) and so it is enough to prove
that CycM(x) ⊆ 〈x〉. On the contrary let y ∈ CycM(x) \ 〈x〉 and hence 〈y, x〉 is cyclic. If y is a
p-element, then we know that 〈y, x〉 has only one subgroup of order p and so 〈x〉 = 〈y〉, which is
a contradiction. Therefore y is not a p-element. So we have an element of order po(y), which is

a contradiction by the structure of Γ (M). So p = |〈x〉| = |CycM(x)|. Therefore |Cyc(G)| divides
p − 1 and p − 1 divides |M|. We know that |Cyc(G)| divides |M| − 1 and so |Cyc(G)| = 1 and

|G| = |M|. Now using Lemma 2.13 we conclude that πe(G) = πe(M) and by Corollary 3.10 the

proof is complete. 2

Remark 3.12. We note that in the main theorem of [5] it is proved that PGL(2, q) is uniquely

determined by πe(G).

Received: February 2009. Revised: August 2010.

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	Introduction