A new characterization of projective special linear groups L₃(q)
In this paper, we prove that projective special linear groups L₃(q), where 0 < q = 5k ± 2 (k ∊ Z) and q² + q + 1 is a prime number can be uniquely determined by their order and the number of elements with same order.
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irk-123456789-1887072023-03-12T01:28:25Z A new characterization of projective special linear groups L₃(q) Ebrahimzadeh, B. In this paper, we prove that projective special linear groups L₃(q), where 0 < q = 5k ± 2 (k ∊ Z) and q² + q + 1 is a prime number can be uniquely determined by their order and the number of elements with same order. 2021 Article A new characterization of projective special linear groups L₃(q) / B. Ebrahimzadeh // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 212–218. — Бібліогр.: 16 назв. — англ. 1726-3255 DOI:10.12958/adm1235 2020 MSC: 20D06, 20D60. http://dspace.nbuv.gov.ua/handle/123456789/188707 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, we prove that projective special linear groups L₃(q), where 0 < q = 5k ± 2 (k ∊ Z) and q² + q + 1 is a prime number can be uniquely determined by their order and the number of elements with same order. |
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Ebrahimzadeh, B. A new characterization of projective special linear groups L₃(q) Algebra and Discrete Mathematics |
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Ebrahimzadeh, B. |
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Ebrahimzadeh, B. |
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A new characterization of projective special linear groups L₃(q) |
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A new characterization of projective special linear groups L₃(q) |
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A new characterization of projective special linear groups L₃(q) |
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A new characterization of projective special linear groups L₃(q) |
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A new characterization of projective special linear groups L₃(q) |
| title_sort |
new characterization of projective special linear groups l₃(q) |
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Інститут прикладної математики і механіки НАН України |
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A new characterization of projective special linear groups L₃(q) / B. Ebrahimzadeh // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 212–218. — Бібліогр.: 16 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT ebrahimzadehb anewcharacterizationofprojectivespeciallineargroupsl3q AT ebrahimzadehb newcharacterizationofprojectivespeciallineargroupsl3q |
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2025-07-16T10:53:33Z |
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2025-07-16T10:53:33Z |
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1837800582682771456 |
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“adm-n2” — 2021/7/19 — 10:26 — page 212 — #48
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 2, pp. 212–218
DOI:10.12958/adm1235
A new characterization of projective special
linear groups L3(q)
B. Ebrahimzadeh
Communicated by A. Yu. Olshanskii
Abstract. In this paper, we prove that projective special
linear groups L3(q), where 0 < q = 5k± 2 (k ∈ Z) and q2 + q + 1 is
a prime number can be uniquely determined by their order and the
number of elements with same order.
1. Introduction
Let G be a finite group, π(G) be the set of prime divisors of the order
of G and πe(G) be the set of the order of elements in G. If k ∈ πe(G), then
we denote the number of elements of order k in G by mk(G) and the set
of the numbers of elements with the same order in G by nse(G). In other
words, nse(G) = {mk(G) : k ∈ πe(G)}. Also we denote a Sylow p-subgroup
of G by Gp and the number of Sylow p-subgroups of G by np(G). The
prime graph Γ(G) of group G is a graph whose vertex set is π(G), and two
vertices u and v are adjacent if and only if uv ∈ πe(G). Moreover, assume
that Γ(G) has t(G) connected components πi, for i = 1, 2, . . . , t(G). In
the case where G is of even order, we assume that 2 ∈ π1.
The characterization of groups by nse(G) pertains to Thompson’s
problem (Problem 12.37 in [10]) which Shi posed in [13]. The first time,
this type of characterization was studied by Shao and Shi. In [12], they
proved that if S is a simple K4- group,then S is characterizable by nse(S)
and |S|. Following this result, in [2–4,7,8, 11], it is proved that sporadic
2020 MSC: 20D06, 20D60.
Key words and phrases: element orders, the number of elements with same
order, prime graph, projective special linear group.
“adm-n2” — 2021/7/19 — 10:26 — page 213 — #49
B. Ebrahimzadeh 213
simple groups, projective special linear groups L2(p) and suzuki groups
Sz(q), where q − 1 is a prime number and also Ree groups 2G2(q),where
q±√
3q+1 is a prime number can be uniquely determined by their orders
and nse(G).
In this paper, we prove that projective special linear groups L3(q),
where 0 < q = 5k ± 2 (k ∈ Z) and q2 + q + 1 is a prime number can be
uniquely determined by their orders and the number of elements with
same order of the group. In fact, we prove the following main theorem.
Main Theorem. Let G be a group with |G| = |L3(q)| and nse(G) =
nse(L3(q)), where 0 < q = 5k±2 (k ∈ Z) and q2+q+1 is a prime number.
Then G ∼= L3(q).
2. Notation and preliminaries
Lemma 2.1 ([6]). Let G be a Frobenius group of even order with kernel
K and complement H. Then
(a) t(G) = 2, π(H) and π(K) are vertex sets of the connected components
of Γ(G);
(b) |H| divides |K| − 1;
(c) K is nilpotent.
Definition 2.2. A group G is called a 2-Frobenius group if there is
a normal series 1 E H E K E G such that G/H and K are Frobenius
groups with kernels K/H and H respectively.
Lemma 2.3 ([1]). Let G be a 2-Frobenius group of even order. Then
(a) t(G) = 2, π(H) ∪ π(G/K) = π1 and π(K/H) = π2;
(b) G/K and K/H are cyclic groups satisfying |G/K| divides
|Aut(K/H)|.
Lemma 2.4 ([15]). Let G be a finite group with t(G) > 2. Then one of
the following statements holds:
(a) G is a Frobenius group;
(b) G is a 2-Frobenius group;
(c) G has a normal series 1EH EK EG such that H and G/K are
π1-groups, K/H is a non-abelian simple group, H is a nilpotent
group and |G/K| divides |Out(K/H)|.
Lemma 2.5 ([5]). Let G be a finite group and m be a positive integer
dividing |G|. If Lm(G) = {g ∈ G | gm = 1}, then m | |Lm(G)|.
“adm-n2” — 2021/7/19 — 10:26 — page 214 — #50
214 Characterization of projective linear groups
Lemma 2.6. Let G be a finite group. Then for every i ∈ πe(G), ϕ(i)
divides mi(G), and i divides
∑
j|imj(G). Moreover, if i > 2, then mi(G)
is even.
Proof. By Lemma 2.5, the proof is straightforward.
Lemma 2.7 ([14]). Let G be a non-abelian simple group such that
(5, |G|) = 1. Then G is isomorphic to one of the following groups:
(a) Ln(q), n = 2, 3, q ≡ ±2 (mod 5);
(b) G2(q), q ≡ ±2 (mod 5);
(c) 2A2(q), q ≡ ±2 (mod 5);
(d) 3D4(q), q ≡ ±2 (mod 5);
(e) 2G2(q), q = 32m+1, m > 1.
Lemma 2.8. Let L be the projective special linear groups L3(q), where
p = q2 + q + 1 is a prime number.Then mp(L) = (p− 1)|L|/(3p) and for
every i ∈ πe(L)− {1, p}, p divides mi(L).
Proof. First we know that |L3(q)| = q3(q3−1)(q2−1).Now since |Lp| = p,
it follows that Lp is a cyclic group of order p. Thus mp(L) = ϕ(p)np(L) =
(p − 1)np(L).Now it is enough to show np(L) = |L|/(3p).By the result
in[15] that yields that the prime p is an isolated vertex of Γ(L).Hence
|CL(Lp)| = p and |NL(Lp)| = xp for a natural number x.We know that
NL(Lp)/CL(Lp) embed in Aut(Lp), which implies x | p− 1.Furthermore,
by Sylow’s Theorem, np(L) = |L : NL(Lp)| and np(L) ≡ 1 (mod p).
Therefore p divides |L|/(xp)− 1.Thus q2 + q + 1 divides (q3(q3 − 1)(q2 −
1)/(xp)− 1.It follows that q2 + q + 1 divides q6 − q5 − q4 + q3 − x so we
have p | 3− x and since x | p− 1, we deduce that x = 3, and the proof is
finished. Let r ∈ πe(L)− {1, p}. Since p is an isolated vertex of Γ(L), it
follows that p ∤ r and pr /∈ πe(L). Thus Lp acts fixed point freely on the
set of elements of order r by conjugation and hence |Lp| | mr(L). So we
conclude that p | mr(L).
3. Proof of the main theorem
In this section, we prove the main theorem in the following lemmas.
We denote the projective special linear groups L3(q), by L and prime
number q2 + q + 1 by p. Recall that G is a group with |G| = |L| and
nse(G) = nse(L).
Lemma 3.1. m2(G) = m2(L), mp(G) = mp(L), np(G) = np(L), p is an
isolated vertex of Γ(G) and p | mk(G) for every k ∈ πe(G)− {1, p}.
“adm-n2” — 2021/7/19 — 10:26 — page 215 — #51
B. Ebrahimzadeh 215
Proof. By Lemma 2.6 if r=2 then m2(G) = φ(2)n2(G) = |G|/6 = |L|/6 =
m2(L).Thus it follows that m2(G) = m2(L).According to Lemma 2.6,
(mp(G), p) = 1. Thus p ∤ mp(G) and hence Lemma 2.8 implies that
mp(G) ∈ {m1(L),m2(L),mp(L)}. Moreover, mp(G) is even, so we deduce
that mp(G) = mp(L). Since Gp and Lp are cyclic groups of order p and
mp(G) = mp(L), we deduce that mp(G) = ϕ(p)np(G) = ϕ(p)np(L) =
mp(L), so np(G) = np(L).
Now we prove that p is an isolated vertex of Γ(G).Assume the con-
trary.Then there is t ∈ π(G) − {p} such that tp ∈ πe(G). So mtp(G) =
ϕ(tp)np(G)k, where k is the number of cyclic subgroups of order t in
CG(Gp). Since np(G) = np(L), it follows that mtp(G) = (t − 1)(p −
1)|L|k/(3p). If mtp(G) = mp(L), then t = 2 and k = 1. Furthermore,
Lemma 2.6 since p > 2 so mp(G) is even. The other hand p | 1 +mp(G).
Now by Lemma 2.5 p | mp(G) so p | 1 +m2(G). Since mp(G) = m2p(G)
so p | 1 + m2p(G) yields p | m2(G) + m2p(G). Since m2(G) = m2(L)
and p | m2(L), we deduce that p | m2p(G) which is a contradiction. So
Lemma 2.8 implies that p | mtp(G). Hence p | t−1 and since mtp(G) < |G|,
we deduce that p− 1 6 3. But this is impossible because p = q2 + q + 1.
Let k ∈ πe(G) − {1, p}. Since p is an isolate vertex of Γ(G), p ∤ k
and pk /∈ πe(G).Thus Gp acts fixed point freely on the set of elements
of order k by conjugation and hence |Gp| | mk(G). So we conclude that
p | mk(G).
Lemma 3.2. The group G is neither a Frobenius group nor a 2-Frobenius
group.
Proof. Let G be a Frobenius group with kernel K and complement H.
Then by Lemma 2.1, t(G) = 2 and π(H) and π(K) are vertex sets of
the connected components of Γ(G) and |H| divides |K| − 1. Now by
Lemma 3.1, p is an isolate vertex of Γ(G). It follows that (i) |H| = p and
|K| = |G|/p or (ii) |H| = |G|/p and |K| = p. Since |H| divides |K|−1, we
deduce that the case (i) can not occur. So |H| = p and |K| = |G|/p, hence
q2 + q + 1 | q3(q3−1)(q2−1)
q2+q+1
− 1, so we have q2 + q + 1 | (q8−q6−q5−q3)
q2+q+1
− 1 in
the way we conclude q2 + q + 1 | (q2 + q + 1)(q4 − 2q3 + 3q − 3) + 2) .
Thus p | 2 which is impossible.
We now show that G is not a 2-Frobenius group. Let G be a 2-
Frobenius group. Then G has a normal series 1EH EK EG such that
G/H and K are Frobenius groups by kernels K/H and H respectively.
Set |G/K| = x. Since p is an isolated vertex of Γ(G), we have |K/H| = p
and |H| = |G|/(xp). By Lemma 2.3, |G/K| divides |Aut(K/H)|. Thus
“adm-n2” — 2021/7/19 — 10:26 — page 216 — #52
216 Characterization of projective linear groups
x | p− 1 and since (q2 + q, q2 + q + 1) = 1, now we deduce q2 + q + 1|H|.
Therefore Ht⋊K/H is a Frobenius group with kernel Ht and complement
K/H, where t = q2 + q + 1. So |K/H| divides |Ht| − 1. It implies that
q2 + q + 1 | (q2 + q + 1)− 1, but this is a contradiction.
Lemma 3.3. The group G is isomorphic to the group L.
Proof. By Lemma 3.1, p is an isolated vertex of Γ(G). Thus t(G) > 1 and
G satisfies one of the cases of Lemma 2.4. Now Lemma 3.2 implies that
G is neither a Frobenius group nor a 2-Frobenius group. Thus only the
case (c) of Lemma 2.4 occurs. So G has a normal series 1EH EK EG
such that H and G/K are π1-groups, K/H is a non-abelian simple group.
Since p is an isolated vertex of Γ(G), we have p | |K/H|. On the other
hand, we know that 5 ∤ |G|.Thus K/H is isomorphic to one of the groups
in Lemma 2.7. Hence we consider the following isomorphisms.
(1) If K/H ∼= L2(q
′), where q′ ≡ ±2 (mod 5), then by(table Id) [15],
π(L2(q
′) = q′, (q′ ± 1)/d, where d = (2, q′ − 1). We assume d = 1, so p, =
q′±1. Now we consider q2+q+1 = q′±1.The first case if q2+q+1 = q′+1
then q(q+1) = q′, that is a contradiction, because q′ = p′m. The second case
if q2+q+1 = q′−1 then we deduce q2+q+2 = q′,now since |L2(q
′)| ∤ |G|,
hence we have a contradiction. Now if d = 2 then p, = q′, (q′ ± 1)/2 so
q2 + q+ 1 = q′, since |L2(q
′)| ∤ |G|, so we have a contradiction. In the way
if q2 + q + 1 = (q′ + 1)/2 then 2q2 + 2q + 1 = q′ since |L2(q
′)| ∤ |G|, hence
we have a contradiction. For other case if q2 + q + 1 = (q′ − 1)/2 then
2q2 + 2q + 3 = q′. But |L2(q
′)| ∤ |G|, so we have a contradiction.
(2) If K/H ∼= G2(q
′), where q′ ≡ ±2 (mod 5), then by(table Ic) [15],
π(G2(q
′) = q′2±q′+1. Thus if q2+q+1 = q′2+q′+1 then q(q+1) = q′(q′+1)
in conclude q = q′. The other hand we know that |K/H| must be divided
|G|, but we can see easily that q6(q6−1)(q2−1) ∤ q3(q3−1)(q2−1), hence
we have a contradiction. Now we consider q2 + q + 1 = q′2 − q′ + 1, since
(q′, q′−1) = 1 we deduce q′ = q+1. The other hand we know |G2(q
′)| must
be divided |G|. But we can see easily that (q+1)6(q+1)6−1)(q+1)2−1) ∤
q3(q3 − 1)(q2 − 1) that is a contradiction.
(3) If K/H ∼= U3(q
′), where q′ ≡ ±2 (mod 5), then by(table Ic) [15],
π(U3(q
′)) = (q′3 − 1)/(q′ + 1)(3, q′ + 1).First if (3, q′ + 1) = 1, then we
consider q2 + q + 1 = q′2 − q′ + 1, hence q(q + 1) = q′(q′ − 1), in conclude
q′ = q + 1. We know that |U3(q
′)| must be divided |G|, but we can see
easily that (q + 1)3(q + 1)3 + 1)(q + 1)2 − 1) ∤ q3(q3 − 1)(q2 − 1) that is
a contradiction. For the other case we have a contradiction, similarily.
(4) If K/H ∼= 2G2(q
′), where q′ = 32m+1, then by(table Id) [15],
π(2G2(q
′)) = q′ ±
√
3q′ + 1. First we consider q2 + q + 1 = q′ ±
√
3q′ + 1,
“adm-n2” — 2021/7/19 — 10:26 — page 217 — #53
B. Ebrahimzadeh 217
in conclude 3m+1(3m ± 1) = q(q + 1),since (q, q + 1) = 1 so q + 1 = 3m+1,
q = 3m + 1. In conclude3m+1(3m + 1) = (3m+1 − 1)(3m+1) then we have
3m+1 = 3m+1−1 that is a contradiction. Now if q2+q+1 = q′−
√
3q′+1
then q(q + 1) = 3m+1(3m − 1), since (q, q + 1) = 1 hence q = 3m − 1 and
q + 1 = 3m+1, first if q = 3m − 1, then by attention to the above equation
we have 3m−1(3m) = 3m+1(3m−1), that in finally we deduce 3m = 3m+1,
that this a contradiction. Now for the other case if q + 1 = 3m+1,then
3m+1(3m+1−1) = 3m+1(3m−1), that we deduce 3m+1−1 = 3m−1. That
is a contradiction.
(5) If K/H ∼= 3D4(q
′) then by(table Ic) [15], π(3D4(q
′) = q′4− q′2+1.
Now we consider q2+q+1 = q′4−q′2+1 in conclude q′2(q′2−1) = q(q+1),
by attention to (q, q+1) = 1 so q = q′2 − 1. Now since |3D4(q
′)| ∤ |G| that
is a contradiction.
Hence we deduce that K/H ∼= L3(q
′). As a result |K/H| = L3(q
′).
Since p is an isolated vertex and also p | |K/H|, we consider q2 + q + 1 =
q′2 + q′ + 1. As aresult q = q′ and also since 1EH EK EG, we deduce
that H = 1, G = K ∼= L.
References
[1] G. Y. Chen, About Frobenius groups and 2-Frobenius groups, J. Southwest China
Normal University, 20, no. 5, 1995, 485–487.
[2] B. Ebrahimzadeh, R. Mohammadyari, A new characterization of symplectic group
C2(3
n), Acta et commentationes universitatis tartuensis de mathematica, 23, no. 1,
(2019).
[3] B. Ebrahimzadeh, A. Iranmanesh, H. Parvizi Mosaed, A new characterization of
Ree group 2G2(q) by the order of group and number of elements with same order,
Int. J. Group Theory, 6, no. 4, 2017, pp. 1–6.
[4] B. Ebrahimzadeh, R. Mohammadyari, A new characterization of suzuki groups,
Archivum mathematicum(Brono) Tomus, 55(2019), 17–21.
[5] G. Frobenius, Verallgemeinerung des sylow’schen satzes, Berliner sitz, 1895, 981–
983.
[6] D. Gorenstein, Finite groups, Harper and Row, New York, 1980.
[7] A. Khalili Asboei, A. Iranmanesh, Characterization of the linear groups L2(p),
Czechoslovak Mathematical Journal, 64, no. 139, 2014, 459–464.
[8] A. Khalili Asboei, S. S. Amiri, A. Iranmanesh, A. Tehranian, A new characterization
of sporadic simple groups by NSE and order, J. Algebra Appl., 12, no.2, 2013,
1250158.
[9] A. S. Kondrat’ev, Prime graph components of finite simple groups, Mathematics
of the USSR-Sbornik, 67, no. 1, 1990, 235–247.
[10] V. D. Mazurov, E. I. Khukhro, eds., Unsolved problems in group theory, The
Kourovka Notebook, 16 ed. Inst. Mat. Sibirsk. Otdel. Akad. Novosibirsk (2006).
“adm-n2” — 2021/7/19 — 10:26 — page 218 — #54
218 Characterization of projective linear groups
[11] H. Parvizi Mosaed, A. Iranmanesh, A. Tehranian, Characterization of suzuki group
by nse and order of group, Bull. Korean Math. Soc., 53, no. 3, 2016, 651–656.
[12] C. G. Shao, W. Shi, Q. H. Jiang, Characterization of simple K4-groups, Front
Math China, 3, no. 3, 2008, 355–370.
[13] W. J. Shi, A new characterization of the Sporadic simple groups, J. Group Theory,
Proc. of the 1987 Singapore Conf., Walter de Gruyter, Berlin, 1989, 531–540.
[14] W. J. Shi, A characterization of U3(2
n) by their element orders J. Southwest-China
Normal Univ, 25, no. 4, 2000, 353–360.
[15] J.S. Williams, Prime graph components of finite groups, J. Algebra, 69, no. 2,
1981, 487–513.
[16] A.V. Zavarnitsine, Recognition of the simple groups L3(q) by element orders, J.
Group Theory, 7, no. 1, 2004, 81–97.
Contact information
Behnam
Ebrahimzadeh
University of Applied Science and Technology
(UAST), ITMC Center, Shiraz, Iran
E-Mail(s): behnam.ebrahimzadeh@gmail.com
Received by the editors: 16.08.2018
and in final form 09.05.2020.
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