Characterization of A₁₆ by a noncommuting graph
Let G be a finite non-Abelian group. We define a graph Γ G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite n...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2010
|
| Назва видання: | Український математичний журнал |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/166289 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Characterization of A₁₆ by a noncommuting graph / M. Davoudi Monfared, M.R. Darafsheh // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1443–1450. — Бібліогр.: 12 назв. — укр. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-166289 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1662892020-02-19T01:27:34Z Characterization of A₁₆ by a noncommuting graph Davoudi Monfared, М. Darafsheh, M.R. Статті Let G be a finite non-Abelian group. We define a graph Γ G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ S ≅ Γ G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except A₁₀, L₄(8), L₄(4), and U₄(4). In this paper, we prove that if A₁₆ denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism ΓA₁₆≅ΓG implies that A₁₆≅G. Нехай G — скінченна неабелівська група. Граф ΓG , який називається непереставним графом групи G, визначено за допомогою множини вершин G—Z(G) таких, що дві вершини x та y є суміжними тоді і тільки тоді, коли xy≠yx. А. Абдоллахі, С. Акбарі та Г. Р. Маймані висунули наступну гіпотезу — ААМ гіпотезу: якщо S є скінченною неабелевою простою групою і G є групою такою, що ΓS≅ΓG, то S≅G. Досі залишається невідомим, чи справджується ця гіпотеза для всіх простих скінченних груп зі зв'язними простими графами, окрім A₁₀, L₄(8), L₄(4) та U₄(4). У статті доведено, що якщо A₁₆ позначає знакозмінну групу степеня 16, то для будь-якої скінченної групи G з ізоморфізму графів ΓA₁₆≅ΓG випливає A₁₆≅G. 2010 Article Characterization of A₁₆ by a noncommuting graph / M. Davoudi Monfared, M.R. Darafsheh // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1443–1450. — Бібліогр.: 12 назв. — укр. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166289 512.5 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Davoudi Monfared, М. Darafsheh, M.R. Characterization of A₁₆ by a noncommuting graph Український математичний журнал |
| description |
Let G be a finite non-Abelian group. We define a graph Γ G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ S ≅ Γ G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except A₁₀, L₄(8), L₄(4), and U₄(4). In this paper, we prove that if A₁₆ denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism ΓA₁₆≅ΓG implies that A₁₆≅G. |
| format |
Article |
| author |
Davoudi Monfared, М. Darafsheh, M.R. |
| author_facet |
Davoudi Monfared, М. Darafsheh, M.R. |
| author_sort |
Davoudi Monfared, М. |
| title |
Characterization of A₁₆ by a noncommuting graph |
| title_short |
Characterization of A₁₆ by a noncommuting graph |
| title_full |
Characterization of A₁₆ by a noncommuting graph |
| title_fullStr |
Characterization of A₁₆ by a noncommuting graph |
| title_full_unstemmed |
Characterization of A₁₆ by a noncommuting graph |
| title_sort |
characterization of a₁₆ by a noncommuting graph |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166289 |
| citation_txt |
Characterization of A₁₆ by a noncommuting graph / M. Davoudi Monfared, M.R. Darafsheh // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1443–1450. — Бібліогр.: 12 назв. — укр. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT davoudimonfaredm characterizationofa16byanoncommutinggraph AT darafshehmr characterizationofa16byanoncommutinggraph |
| first_indexed |
2025-07-14T21:07:12Z |
| last_indexed |
2025-07-14T21:07:12Z |
| _version_ |
1837657995481186304 |
| fulltext |
UDC 512.5
M. R. Darafsheh (School Math., College Sci., Univ. Tehran, Iran),
M. Davoudi Monfared (Islamic Azad Univ., Tafresh, Iran)
CHARACTERIZATION OF A16 BY NON-COMMUTING GRAPH
ХАРАКТЕРИЗАЦIЯ A16 НЕПЕРЕСТАВНИМ ГРАФОМ
Let G be a finite non-Abelian group. We define a graph ΓG, called the non-commuting graph of G, with vertex
set G−Z(G) such that two vertices x and y are adjacent if and only if xy 6= yx. A. Abdollahi, S. Akbari and
H. R. Maimani put forward a conjecture, the AAM’s Conjecture as follows: If S is a finite non-Abelian simple
group and G is a group such that ΓS
∼= ΓG, then S ∼= G. It is still unknown if this conjecture holds for all
simple finite groups with connected prime graph except A10, L4(8), L4(4) and U4(4). In this paper we prove
that if A16 denotes the alternating group of degree 16, then for any finite group G, the graph isomorphism
ΓA16
∼= ΓG implies A16
∼= G.
Нехай G — скiнченна неабелiвська група. Граф ΓG, який називається непереставним графом групи G,
визначено за допомогою множини вершин G − Z(G) таких, що двi вершини x та y є сумiжними тодi
i тiльки тодi, коли xy 6= yx. А. Абдоллахi, С. Акбарi та Г. Р. Майманi висунули наступну гiпотезу —
ААМ гiпотезу: якщо S є скiнченною неабелевою простою групою i G є групою такою, що ΓS
∼= ΓG,
то S ∼= G. Досi залишається невiдомим, чи справджується ця гiпотеза для всiх простих скiнченних
груп зi зв’язними простими графами, окрiм A10, L4(8), L4(4) та U4(4). У статтi доведено, що якщо
A16 позначає знакозмiнну групу степеня 16, то для будь-якої скiнченної групи G з iзоморфiзму графiв
ΓA16
∼= ΓG випливає A16
∼= G.
1. Introduction. The study of relation between groups and graphs is one of the main
research topic in group theory. There are several ways to associate a graph to a group
G. The graph we will consider in this paper is denoted by ΓG and is called the non-
commuting graph of G. The vertex set of ΓG is V (ΓG) = G−Z(G) where Z(G) is the
center of G and two distinct vertices x and y are joined whenever xy 6= yx. It is clear
that if G is abelian, then ΓG is the null graph. Hence in what follows we will assume
that G is a non-Abelian group. Another graph associated to a finite group G is the prime
graph GK(G) introduced by Gruenberg – Kegel. The vertex set of GK(G) is π(G), the
set of all the prime divisors of the order of G. Two distinct primes p and q are adjacent
if and only if G contains an element of order pq.
For a graph X, we denote the set of vertices and edges of X by V (X) and E(X)
respectively. Two graphs X and Y are isomorphic and we denote it by X ∼= Y, if there
exists a bijective map φ : V (X) −→ V (Y ) such that if x and y are adjacent in X, then
φ(x) and φ(y) are adjacent in Y and vice-versa. For a group G, we denote by k(G)
the number of conjugacy classes of G and N(G) = {n ∈ N|G has a conjugacy class C
such that |C| = n}. Also ClG(g) denotes the conjugacy class containing g ∈ G.
In [1] relation between some graph theoretical properties of ΓG and the group theory
properties of the group G are studied. In particular the following two conjectures are
raised.
Conjecture 1. Let G be a finite non-Abelian group. If there is a group such that
ΓG ∼= ΓH , then |G| = |H|.
c© M. R. DARAFSHEH, M. DAVOUDI MONFARED, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11 1443
1444 M. R. DARAFSHEH, M. DAVOUDI MONFARED
Conjecture 2. Let S be a finite non-Abelian simple group. If G is a group such
that ΓG ∼= ΓS , then G ∼= S.
There are many articles dealing with the characterization of simple groups by its
non-commuting graph. In [3], M. R. Darafsheh proved Conjecture 1 for any simple
group G. Also if GK(G) is a non-connected graph, Conjecture 2 is verified for many
simple groups. In [6], A. Iranmanesh and J. Jafarzadeh verified Conjecture 2 when G
and S are both simple groups. In [10], L. Wang and W. Shi verified Conjecture 2 for
S ∼= L2(q). But if GK(G) is a connected graph the structure theorem for the group
G does not work in the general case and the problem of characterizing the group G
via its non-commuting graph becomes difficult. In the case that GK(G) is a connected
graph, the following partial results have been obtained so far. In [11], L. Wang and
W. Shi verified Conjecture 2 For S ∼= A10. In [12], L. Zhang and W. Shi proved that
Conjecture 2 is true for L4(8). In [4], Conjecture 2 is verified for the groups L4(4) and
U4(4). The groups A10, L4(8), L4(4) and U4(4) have connected prime graphs. Our aim
in this paper is to verify the above Conjecture for the alternating group of degree 16,
A16, that has connected prime graph. In fact, we will prove the following theorem.
Theorem 1. Let G be a finite group such that ΓG ∼= ΓA16
, then G ∼= A16.
2. Preliminaries. In this section we list some basic and known results which will
be used in proving Theorem 1.
Lemma 1 [7, p. 98]. If |G| = pqr, where p, q and r are distinct primes, then G is
not simple.
Lemma 2 (Lemma 3.27 of [1]). If G is a finite group, then 2|E(ΓG)| = |G|(|G| −
− k(G)).
Theorem 2 (P. Hall [8, p. 108]). If G is a solvable group of order mn where (m,
n) = 1, then G contains a subgroup of order m. Moreover any two subgroup of order
m are conjugate.
Denote by t(G) the maximal number of primes in π(G) which are pairwise nonadja-
cent in GK(G). Also we denote by t(2, G) the maximal number of vertices containing 2
but pairwise nonadjacent in GK(G). t(G) is called the independence number of GK(G)
and t(2, G) is called the 2-independence number of the graph GK(G).
Theorem 3 [9]. Let G be a finite group satisfying the two conditions:
(a) there exist three primes in π(G) pairwise nonadjacent in GK(G), i.e., t(G) ≥ 3;
(b) there exist an odd prime in π(G) nonadjacent to 2 in GK(G), i.e., t(2, G) ≥ 2.
Then there is a finite non-Abelian simple group S such that S ≤ G = G/K ≤
≤ Aut(S) for the maximal normal solvable subgroup K of G. Furthermore, t(S) ≥
≥ t(G)− 1 and one of the following statements holds:
(1) S ∼= A7 or PSL(2, q) for some odd q, and t(S) = t(2, S) = 3;
(2) for every prime p ∈ π(G) nonadjacent to 2 in GK(G) a Sylow p-subgroup of G
is isomorphic to a Sylow p-subgroup of S. In particular, t(2, S) ≥ t(2, G).
3. Characterization of A16 by non-commuting graph. We know that |A16| =
= 10461394944000 = 214 · 36 · 53 · 72 · 11 · 13 and by [5] it has 123 conjugacy classes.
For proving Theorem 1, we need the size of conjugacy classes, centralizer orders, order
of elements and the number of conjugacy classes in A16. We omit the details of these
items and refer to [5] for some of them in the following.
Lemma 3. Let G be a finite group such that ΓG ∼= ΓA16
, then
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
CHARACTERIZATION OF A16 BY NON-COMMUTING GRAPH 1445
(1) |G| = |A16|,
(2) k(G) = k(A16),
(3) if φ : ΓG −→ ΓA16
is a graph isomorphism, then |CG(g)| = |CA16
(φ(g))| and
|ClG(g)| = |ClA16(φ(g))| for all g ∈ G− Z(G). In particular N(G) = N(A16).
Proof. By [3] we have |G| = |A16| and proof of (1) is immediate. Also from
Lemma 2 we know that 2|E(ΓG)| = |G|(|G| − k(G)) and by ΓG ∼= ΓA16 and (1) we
obtain (2). It is clear that deg(g) = |G| − |CG(g)| for every g ∈ G − Z(G) where
deg(g) denotes the degree of g in the graph ΓG. From the graph isomorphism, we have
deg(g) = deg(φ(g)) and so |G| − |CG(g)| = |A16| − |CA16(φ(g))|. Hence from (1) we
have |CG(g)| = |CA16
(φ(g))| and |ClG(g)| = |ClA16
(φ(g))| where ClG(g) denotes the
conjugacy class containing g. Finally from the equality of sizes of conjugacy classes,
we obtain N(G) = N(A16).
Lemma 3 is proved.
In GK(A16), from the set of orders of elements, we know that 2, 3, 5 and 7 are
adjacent to each other, 11 is adjacent to 2, 3 and 5 and finally 13 is adjacent to 3. In the
following we draw GK(A16) (Fig. 1).
we obtain (2). It is clear that deg(g) = |G|− |CG(g)| for every g ∈ G−Z(G)
where deg(g) denotes the degree of g in the graph ΓG. From the graph
isomorphism, we have deg(g) = deg(φ(g)) and so |G| − |CG(g)| = |A16| −
|CA16(φ(g))|. Hence from (1) we have |CG(g)| = |CA16(φ(g))| and |ClG(g)| =
|ClA16(φ(g))| where ClG(g) denotes the conjugacy class containing g. Finally
from the equality of sizes of conjugacy classes, we obtain N(G) = N(A16).
In GK(A16), from the set of orders of elements, we know that 2, 3, 5
and 7 are adjacent to each other, 11 is adjacent to 2, 3 and 5 and finally 13
is adjacent to 3. In the following we draw GK(A16).
✉
✉
✉
✉
✉
✉
2
3
5
7
11
13
Figure 1
In the next lemma, we show that if the non-commuting graphs of G and
A16 are isomorphic, then their prime graphs are equal.
Lemma 4 Let G be a finite group such that ΓG
∼= ΓA16, then GK(G) =
GK(A16).
Proof. By lemma 3, G and A16 have the same set of centralizer orders
and π(G) = π(A16) = {2, 3, 5, 7, 11, 13}. By [5] we know that A16 has a
centralizer order |CA16(a)| = 24.33 for a ∈ A16. So G has a centralizer order
5
Fig. 1
In the next lemma, we show that if the non-commuting graphs of G and A16 are
isomorphic, then their prime graphs are equal.
Lemma 4. Let G be a finite group such that ΓG ∼= ΓA16
, then GK(G) =
= GK(A16).
Proof. By Lemma 3, G and A16 have the same set of centralizer orders and π(G) =
= π(A16) = {2, 3, 5, 7, 11, 13}. By [5] we know that A16 has a centralizer order
|CA16(a)| = 24 · 33 for a ∈ A16. So G has a centralizer order |CG(g)| = 24 · 33
where φ(g) = a in the graph isomorphism φ : ΓG → ΓA16
. If o(g) = 2α, 1 ≤ α ≤ 4,
then g commutes with an element of order 3 and so G has an element of order 6. If
o(g) = 2α · 3β , 1 ≤ α ≤ 4 and 1 ≤ β ≤ 3, then G has an element of order 6. If
o(g) = 3β , 1 ≤ β ≤ 3, then g commutes with an element of order 2 and so G has an el-
ement of order 6. Thus in any case G has an element of order 6 and 2 is adjacent to 3 in
GK(G). By [5] and using similar argument we obtain G has elements g1, g2, g3, g4, g5,
g6, g7 and g8 with |CG(g1)| = 3·5, |CG(g2)| = 2·7, |CG(g3)| = 3·13, CG(g4) = 3·11,
|CG(g5)| = 5 · 11, |CG(g6)| = 22 · 11, |CG(g7)| = 23 · 5 and |CG(g8)| = 32 · 7· So 2, 3
and 5 are adjacent to each other, 7 is adjacent to 2 and 3, 11 is adjacent to 2, 3, 5 and
13 is adjacent to 3. Next we show that 5 is adjacent to 7. From centralizer orders of A16
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1446 M. R. DARAFSHEH, M. DAVOUDI MONFARED
and by Lemma 3, we deduce that G has a centralizer order |CG(g)| = 3 · 5 · 7. Now by
Lemma 1 and Theorem 2, G has a subgroup K with |K| = 5 · 7 = 35. But every group
of order 35 is Abelian (and cyclic). So G has an element h with o(h) = 35. Thus 5 is
adjacent to 7 in GK(G).
Next we prove that 2 is not adjacent to 13. If 2 is adjacent to 13, then G has an
element g with o(g) = 26. It implies that 26
∣∣|CG(g)|. From Lemma 3 and centralizer
orders of A16 taken from [5] we obtain |CG(g)| = 29 · 36 · 52 · 7 · 11 · 13. Now by
o(g) = 26 and |CG(g)|, G has elements of order 2, 3, 5, 7, and 11 such that both 2 and
13 divide centralizer order of these elements. So G has at least five conjugacy classes
of elements with different orders where both 2 and 13 divide the centralizer order of
each element. But by Lemma 3 and [5], G has only one conjugacy class of element with
centralizer order divisible by 26, a contradiction. Similarly 13 is not adjacent to 5, 7, and
11. We will prove that 7 is not adjacent to 11 in GK(G). If this happens, then G has an
element h with o(h) = 77. So 77
∣∣|CG(h)| and from Lemma 3 and centralizer orders of
A16, we have |CG(h)| = 27 · 34 · 5 · 7 · 11, 29 · 36 · 52 · 7 · 11 · 13 or 212 · 35 · 52 · 7 · 11.
In these cases G has elements of order 2, 3, 5, 7 and 11 such that 77 divides centralizer
order of these elements. But G has only three centralizer orders divisible by 77. This is
a contradiction and 7 is not adjacent to 11 in GK(G). Therefore GK(G) = GK(A16)
and the proof is completed.
Lemma 4 is proved.
Now by Theorem 3, there is a non-Abelian simple group S such that S ≤ G/K ≤
≤ Aut(S) for the maximal normal solvable subgroup K of G. Furthermore, t(S) ≥
≥ t(G)− 1 and one of the following statements holds:
(1) S ∼= A7 or PSL(2, q) for some odd q, and t(S) = t(2, S) = 3,
(2) for every prime p ∈ π(G) nonadjacent to 2 in GK(G) a Sylow p-subgroup of G
is isomorphic to a Sylow p-subgroup of S. In particular, t(2, S) ≥ t(2, G).
In the next lemma we will prove that condition (1) of Theorem 3 does not hold. So
condition (2) holds.
Lemma 5. The non-Abelian simple group S in Theorem 5 has a Sylow 13-
subgroup of order 13.
Proof. We show that conclusion (1) of Theorem 5 does not hold. If S ∼= A7, then
|S| = 7!/2 and |Aut(S)| = 7!. So 7!/2 ≤ |G/K| ≤ 7! and since |G| = 16!/2 we
obtain 4151347200 ≤ |K| ≤ 8302694400. So |K| = 4151347200 or 8302694400 and
in any case 11 · 13
∣∣|K|. Since K is solvable, by Theorem 2 K has a subgroup K1 of
order 11 · 13. By Sylow’s theorems, every group of order 11 · 13 is cyclic. Hence G
has an element of order 143 that implies 11 is adjacent to 13 in GK(G) which is a
contradiction by Fig. 1. Therefore S 6∼= A7.
If S ∼= PSL(2, q) for some odd q, then by S ≤ G/K, |PSL(2, q)| = 1
(n, q − 1)
q(q2−
− 1) and comparing q in |G| and |PSL(2, q)|, the following cases are raised: S ∼=
∼= PSL(2, 3n), 2 ≤ n ≤ 6, PSL(2, 5), PSL(2, 52), PSL(2, 53), PSL(2, 7),
PSL(2, 72), PSL(2, 11) or PSL(2, 13).
If S ∼= PSL(2, 32), then |S| = 24 · 32 · 5 and |Aut(S)| = 25 · 32 · 5. So 24 · 32 · 5 ≤
≤ |G/K| ≤ 25 · 32 · 5. From |G| we deduce that 29 · 34 · 52 · 72 · 11 · 13 ≤ |K| ≤
≤ 210 · 34 · 52 · 72 · 11 · 13. Since |K|
∣∣|G| we obtain |K| = 29 · 34 · 52 · 72 · 11 · 13 or
|K| = 210 · 34 · 52 · 72 · 11 · 13. In any case by Theorem 2, K has a subgroup of order
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
CHARACTERIZATION OF A16 BY NON-COMMUTING GRAPH 1447
11 · 13. Every group of order 11 · 13 is Abelian (and cyclic) which implies that 11 is
adjacent to 13 in GK(G) and this contradicts Lemma 4. If S ∼= PSL(2, 33), then by
similar argument we obtain 7 is adjacent to 11 in GK(G) and this is a contradiction.
If S ∼= PSL(2, 34), then |S| = 265680 = 24 · 34 · 5 · 41. By Theorem 3 we know
that |S|
∣∣|G|. So 41
∣∣G| and this contradicts 41 - |G|. If S ∼= PSL(2, 35), then 61
∣∣|S|
and this contradicts 61 - |G|. If S ∼= PSL(2, 36), then 73
∣∣|S| and this contradicts
73 - |G|. If S ∼= PSL(2, 5), then similar to the case S ∼= PSL(2, 32), we deduce
that G has a subgroup of order 11 · 13. Thus 11 is adjacent to 13 in GK(G) and this
contradicts GK(G) = GK(A16). If S ∼= PSL(2, 52), then |S| = 23 · 3 · 52 · 13 and
|Aut(S)| = 25 · 3 · 52 · 13. So 29 · 35 · 5 · 72 · 11 ≤ |K| ≤ 211 · 35 · 5 · 72 · 11 and
possibilities for |K| are |K| = 29 · 35 · 5 · 72 · 11, 210 · 35 · 5 · 72 · 11, 29 · 36 · 5 · 72 · 11 or
211 ·35 ·5 ·72 ·11. In any case by Theorem 2, K has a subgroup K1 with |K1| = 11 ·72.
If P is an 11-Sylow subgroup of K1, then by Sylow’s theorms P is normal in K1. If
t ∈ K1 with o(t) = 7, then P 〈t〉 is a subgroup of K1 of order 77. Every group of
order 77 is Abelian (and cyclic) implying that 7 is adjacent to 11 in GK(G) and this
contradicts GK(G) = GK(A16). If S ∼= PSL(2, 53), then 31
∣∣|S| and this contradicts
31 - |G|. If S ∼= PSL(2, 7) or PSL(2, 72), then similar to the case S ∼= PSL(2, 32)
we have 11 is adjacent to 13 in GK(G) which is a contradiction. If S ∼= PSL(2, 11),
then similar to the case S ∼= PSL(2, 33), we obtain 7 is adjacent to 13 in GK(G),
a contradiction. If S ∼= PSL(2, 13), then similarly we show that 7 is adjacent to 11
in GK(G) and this is a contradiction. Therefore conclusion (1) of Theorem 3 does not
hold. So conclusion (2) of Theorem 3 holds and S has a 13-Sylow subgroup of order 13.
Lemma 5 is proved.
3.1. Proof of the main theorem. Now by [2] we consider each of the finite non-
Abelian simple groups as a candidate for S.
(1) S ∼= An for n ≥ 5. By Lemma 5 we know that 13
∣∣|S|, so n ≥ 13. Therefore
S ∼= A13, A14, A15 or A16. If S ∼= A13, then by Theorem 3 we obtain 13!/2 ≤ G/K ≤
13!. Thus 3360 ≤ |K| ≤ 6720 and |K| = 3360 = 25 ·3·5·7 or 6720 = 26 ·3·5·7· On the
other handK is normal inG, so in any caseK contains four distinct conjugacy classes of
elements with orders 2, 3, 5 and 7. But the four smallest orders for conjugacy classes of
G are 1120, 5460, 104832, 320320. Therefore K can not contain four conjugacy classes
which is a contradiction. The cases S ∼= A14 or A15 similarly lead to contradiction. If
S ∼= A16, then from S ≤ G/K we obtain |S| = |G| and |K| = 1. So S = G and in
this case S = G ∼= A16. Thus if S ∼= An for n ≥ 5, then S ∼= A16 and in this case
S = G ∼= A16.
(2) If S is isomorphic to one of the sporadic simple groups, then by Lemma 5 we
have 13
∣∣|S|. Also the possible prime divisors of |S| are 2, 3, 5, 7, 11 or 13, hence we
obtain S ∼= Suz or Fi22. If S ∼= Suz, then |S| = 213·37·52·7·11·13. So 37
∣∣|S| implying
that 37
∣∣|G| and this is a contradiction. If S ∼= Fi22, then |S| = 217 · 39 · 52 · 7 · 11 · 13.
So 217
∣∣|S| implying that 217
∣∣|G|, a contradiction. Therefore S is not one of the sporadic
simple groups.
(3) S is one of the classical groups PSL(n, q) for n ∈ N and prime power q.
Since |PSL(n, q)| =
1
(n, q − 1)
qn(n−1)/2
∏n
i=2
(qi − 1) and S ≤ G/K and |G| =
= 214 · 36 · 53 · 72 · 11 · 13, q must be a power of 2, 3, 5, 7, 11 or 13. In Lemma 5
we showed that S is not one of the classical groups PSL(2, q) for odd q. So S may
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1448 M. R. DARAFSHEH, M. DAVOUDI MONFARED
be isomorphic to one of the following cases: PSL(2, 2n), 2 ≤ n ≤ 14, PSL(3, 2n),
1 ≤ n ≤ 4, PSL(3, 3), PSL(3, 32), PSL(3, 5), PSL(4, 2), PSL(4, 22) or PSL(5, 2).
If S ∼= PSL(2, 2n), 2 ≤ n ≤ 14 and n 6= 6, 12, PSL(3, 2), PSL(3, 22), PSL(3, 23),
PSL(3, 5), PSL(4, 2), PSL(4, 4) or PSL(5, 2), then 13 - |S| and this contradicts
Lemma 5. If S ∼= PSL(2, 26), PSL(3, 3) or PSL(3, 32) then similar to the Lemma 5,
the case S ∼= PSL(2, 32), we deduce that 7 is adjacent to 11 in GK(G) and this is
a contradiction. If S ∼= PSL(2, 212) or PSL(3, 24), then 17
∣∣|S| and this contradicts
17 - |G|. So S is not one of the classical groups PSL(n, q).
(4) S is one of the classical groups PSU(n, q2) for n ∈ N and prime power q.
Similar to the case (3) we obtain that S ∼= PSU(2, 22n), 2 ≤ n ≤ 7, PSU(2, 32),
PSU(2, 34), PSU(2, 36), PSU(2, 52), PSU(2, 72), PSU(3, 22), PSU(3, 24) or
PSU(3, 32). If S ∼= PSU(2, 24), PSU(2, 28), PSU(2, 210), PSU(2, 214), PSU(2, 32),
PSU(2, 34) or PSU(2, 72), then 13 - |S| and this contradicts Lemma 5. If S ∼=
∼= PSU(2, 26), then |S| = 26 · 32 · 5 · 7 · 13 and |Aut(S)| = 27 · 33 · 5 · 7 · 13.
So 26 · 32 · 5 · 7 · 13 ≤ |G/K| ≤ 27 · 33 · 5 · 7 · 13 and by |G| we have 27 · 33 · 52 · 7 · 11 ≤
≤ |K| ≤ 28 · 34 · 52 · 7 · 11. Then the possibilities for |K| are 27 · 33 · 52 · 7 · 11,
28 · 33 · 52 · 7 · 11, 27 · 33 · 53 · 7 · 11, 27 · 34 · 52 · 7 · 11 or 28 · 34 · 52 · 7 · 11. By
Theorem 2, in any case K has a subgroup K1 of order 77. But every group of order
77 is Abelian (and cyclic) which means that 7 is adjacent to 11 in GK(G) and this is
a contradiction. Similar to this argument S can not be isomorphic to either PSU(2, 52)
or PSU(3, 22). If S ∼= PSU(2, 212) or PSU(3, 24), then 17
∣∣|S| and this contradicts
17 - |G|. If S ∼= PSU(2, 36), then 73
∣∣|S| and this contradicts 73 - |G|. Therefore S is
not one of the classical groups PSU(n, q2).
(5) S is one of the classical groups PSP (2l, q) or PΩ(2l+ 1, q). At first we assume
that S ∼= PSP (2l, q). From PSP (2, q) ∼= PSL(2, q) and proof of Lemma 5 and
case (3), we obtain S 6∼= PSP (2, q). Using the fact that |S|
∣∣|G| and |PSP (2l, q)| =
=
1
(2, q − 1)
ql
2
∏l
i=1
(q2i − 1) we obtain S ∼= PSP (4, 2), PSP (4, 22), PSP (4, 23),
PSP (4, 3) or PSP (6, 2). If S ∼= PSP (4, 2), PSP (4, 22), PSP (4, 3) or PSP (6, 2),
then 13 - |S| and this contradicts Lemma 5. If S ∼= PSP (4, 23), then |S| = 212 ·34 ·5·72 ·
·13 and |Aut(S)| = 213 ·35 ·5·72 ·13 and so 212 ·34 ·5·72 ·13 ≤ |G/K| ≤ 213 ·35 ·5·72 ·13.
Thus the possibilities for |K| are 2·3·52·11, 22·3·52·11, 2·3·53·11, 2·32·52·11, 23·3·52·11
or 22 · 32 · 52 · 11. In any case K contains four distinct conjugacy classes of elements
of orders 2, 3, 5 and 11. The first four smallest conjugacy classes of G by Lemma 3
and [5] have orders 1120, 5460, 104832 and 320320. From |K| ≤ 9900 we deduce
that K can not contain four distinct conjugacy classes and this is a contradiction. So S
is not one of the symplectic groups PSP (2l, q). If S is one of the orthogonal groups
PΩ(2l + 1, q), then using PSL(2, q) ∼= PΩ(3, q) and proof of Lemma 5 and case (3)
we get that S 6∼= PΩ(3, q). So S may be isomorphic to one of the following cases:
PΩ(5, 2), PΩ(5, 22), PΩ(5, 23), PΩ(5, 3) or PΩ(7, 2). If S ∼= PΩ(5, 2), PΩ(5, 22),
PΩ(5, 3) or PΩ(7, 2), then 13 - |S| and this contradicts Lemma 5. If S ∼= PΩ(5, 23),
then we obtain |S| = 212 · 34 · 5 · 72 · 13 and |Aut(S)| = 213 · 35 · 5 · 72 · 13. By
similar argument to the case S ∼= PSP (4, 23), we get a contradiction. So S is not one
of the orthogonal groups PΩ(2l + 1, q). Therefore S is not one of the classical groups
PSP (2l, q) or PΩ(2l + 1, q).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
CHARACTERIZATION OF A16 BY NON-COMMUTING GRAPH 1449
(6) S is one of the classical groups PΩε(2l, q) for ε ∈ {+1,−1}. Then using the
order of G and |PΩε(2l, q)|, q can be a power of 2, 3, 5 or 7. So S may be one
of the groups PΩε(4, 2n), 1 ≤ n ≤ 7, PΩε(4, 3), PΩε(4, 32), PΩε(4, 33), PΩε(4, 5),
PΩε(4, 7), PΩε(6, 2), PΩε(6, 22), PΩε(6, 3) or PΩε(8, 2). If S ∼= PΩε(4, 2),
PΩε(4, 22), PΩε(4, 25), PΩε(4, 27), PΩε(4, 3), PΩε(4, 32), PΩε(4, 7), PΩε(6, 2) or
PΩε(8, 2), then 13 - |S| and this contradicts Lemma 5. If S ∼= PΩε(4, 8), PΩε(4, 5),
or PΩε(6, 3), then we consider two cases. If ε = +1, then 13 - |S| and this contradicts
Lemma 5. If ε = −1, then similar to the case (4) for S ∼= PSU(2, 26), we obtain
that 7 is adjacent to 11 in GK(G) and this is a contradiction. If S ∼= PΩε(4, 24) or
PΩε(6, 22), then 17
∣∣|S| and this contradicts 17 - |G|. If S ∼= PΩε(4, 26), then we con-
sider two cases. If ε = +1, then 132
∣∣|S| and this contradicts 132 - |G|. If ε = −1, then
17
∣∣|S| and this contradicts 17 - |G|. So S is not one of the classical groups PΩε(2l, q).
(7) S is one of the exceptional Chevalley groups F4(q), G2(q), E6(q), E7(q), E8(q).
If S ∼= G2(q), then using the orders of |G2(q)| and |G| and the fact that S ≤ G/K,
q may be one of the 2, 3 or 22. If S ∼= G2(2), then 13 - |S| which is a contradiction.
If S ∼= G2(22) or G2(3), then similar to the case (4) for PSU(2, 26), we obtain that 7
is adjacent to 11 in GK(G) which is a contradiction. If S ∼= F4(q), E6(q), E7(q) or
E8(q), then |S| has a factor of form q24, q36, q63 or q120. But from S ≤ G/K we know
that the maximum factor in |S| can be q14 (for q = 2) and this is a contradiction. So S
is not one of the exceptional Chevalley groups.
(8) S is one of the twisted Chevalley groups or Tits group. Then S ∼=2 D4(q),
2F4(22n+1), 2E6(q), 2G2(32n+1), 2G2(3)′, 2B2(22n+1) or T, the Tits group. If S ∼=
∼= 3D4(q), then using S ≤ G/K, |G| and |3D4(q)|, the only possibility for q is 2. If
S ∼=3 D4(2), then |S| = 212 · 34 · 72 · 13 and |Aut(S)| = 212 · 35 · 72 · 13. So from
S ≤ G/K ≤ Aut(S) and |G| we have |K| = 22 ·3 ·53 ·11, 23 ·3 ·53 ·11 or 22 ·32 ·53 ·11.
K is normal in G and in any case has four conjugacy classes of elements of order 2, 3, 5
and 11. But the four smallest conjugacy classes of G (by [5] and Lemma 3) have orders
1120, 5460, 104832 and 320320. From |K| we deduce that K can not contain four
distinct conjugacy classes which is a contradiction. If S ∼= 2F4(22n+1) for n > 0, then
by |2F4(22n+1)| we have 236
∣∣|S| and this contradicts 236 - |G|. If S ∼=2 G2(32n+1) for
n > 0, then 39
∣∣|S| and this contradicts 39 - |G|. If S ∼=2 E6(q) for n > 0, then |S| has
a factor of the form q36. But the largest factor in |S| can be q14 (for q = 2) and this is
a contradiction. If S ∼=2 B2(22n+1) for n > 0, then by |S| we obtain that n may be 1, 2
or 3. If n = 1, then S ∼=2 B2(23). So |S| = 26 ·5 ·7 ·13 and |Aut(S)| = 26 ·3 ·5 ·7 ·13.
Similar to the case (4) for PSU(2, 26) we get a contradiction with 7 adjacent to 11 in
GK(G). If n = 2 or 3, then S ∼=2 B2(25) or 2B2(27). In any case 13 - |S| and this
contradicts Lemma 5. If S ∼=2 G2(3)′, then 13 - |S| and this contradicts Lemma 5. If
S ∼= T, the Tits group, then |S| = 211 ·33 ·52 ·13 and |Aut(S)| = 212 ·33 ·52 ·13. Similar
to the case (4) for PSU(2, 26), we deduce that 7 is adjacent to 11 in GK(G) and this
is a contradiction. So S is not one of the twisted Chevalley groups or Tits group.
Conclusion of the proof of Theorem 1. In cases (1) – (8) we considered S to be
a non-Abelian finite simple group and showed that the only possibility is S ∼= A16. So
|S| = |G| = 214 ·36 ·53 ·72 ·11 ·13 and from S ≤ G/K ≤ Aut(S) for a maximal normal
solvable subgroup K of G, we obtain S = G = A16 and K = {1}. Thus S = G = A16
and the proof is completed.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1450 M. R. DARAFSHEH, M. DAVOUDI MONFARED
1. Abdollahi A., Akbari S., Maimani H. R. Non-commuting graph of a group // J. Algebra. – 2006. – 298.
– P. 468 – 496.
2. Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A. Atlas of finite groups. – Oxford:
Clarendon Press, 1985.
3. Darafsheh M. R. Groups with the same non-commuting graph // Discrete Appl. Math. – 2009. – 157. –
P. 833 – 837.
4. Darafsheh M. R., Monfared M. Davoudi. A characterization of the groups PSU(4, 4) and PSL(4, 4)
by non-commuting graph (submited).
5. The gap group, gap-groups, algoritms and programming, version 4.4, http://www.gap-system.org, 2005.
6. Iranmanesh A., Jafarzadeh A. Characterization of finite groups by their commuting graph // Acta math.
hung. – 2007. – 23. – P. 7 – 13.
7. Rose John S. A course on group theory. – Cambridge: Cambridge Univ. Press, 1978.
8. Rotman J. J. An introduction to the theory of groups. – Fourth ed. – New York: Springer, 1994.
9. Vasilév A. V. On connection between the structure of a finite group and the proprties of its prime graph
// Sib. Math. J. – 2005. – 46, № 3. – P. 396 – 404.
10. Wang L., Shi W. A new characterization of L2(q) by noncommuting graph // Front. Math. China. –
2007. – 2, № 1. – P. 143 – 148.
11. Wang L., Shi W. A new characterization of A10 by its noncommuting graph // Communs Algebra. –
2008. – 36, № 2. – P. 523 – 528.
12. Zhang L., Shi W. Noncommuting graph characterization of some simple groups with connected prime
graph // Int. Electron. J. Algebra. – 2009. – 5. – P. 169 – 181.
Received 10.06.09,
after revision — 14.06.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
|