Characterization of the group $G_2(5)$ by the prime graph
Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma (G)$ with vertex set $\pi (G)$ and the set of all prime divisors of $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $G\Gamma (G) = \Gamma (G_2(5))$, then...
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| author | Darafsheh, M. R. Nosratpour, P. Дарафшех, М. Р. Носратпур, П. |
| author_facet | Darafsheh, M. R. Nosratpour, P. Дарафшех, М. Р. Носратпур, П. |
| author_sort | Darafsheh, M. R. |
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| description | Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma (G)$ with vertex set $\pi (G)$ and the set of all prime divisors
of $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if
$G\Gamma (G) = \Gamma (G_2(5))$, then $G$ has a normal subgroup $N$ such that $\pi (N) \subseteq \{ 2, 3, 5\}$ and $G/N \sim = G_2(5)$. |
| first_indexed | 2026-03-24T02:15:01Z |
| format | Article |
| fulltext |
UDC 512.5
P. Nosratpour (Ilam Branch, Islamic Azad Univ., Ilam, Iran),
M. R. Darafsheh (School Math., Statistics and Comput. Sci., College Sci., Univ. Tehran, Iran)
CHARACTERIZATION OF THE GROUP \bfitG \bftwo (\bffive ) BY THE PRIME GRAPH
ХАРАКТЕРИЗАЦIЯ ГРУПИ \bfitG \bftwo (\bffive ) ЗА ДОПОМОГОЮ ПРОСТОГО ГРАФА
Let G be a finite group. The prime graph of G is a graph \Gamma (G) with vertex set \pi (G) and the set of all prime divisors
of | G| , where two distinct vertices p and q are adjacent by an edge if G has an element of order pq. We prove that if
\Gamma (G) = \Gamma (G2(5)), then G has a normal subgroup N such that \pi (N) \subseteq \{ 2, 3, 5\} and G/N \sim = G2(5).
Нехай G — скiнченна група. Простим графом G називається граф \Gamma (G) з множиною вершин \pi (G) та множиною
всiх простих дiльникiв | G| , в якому двi рiзнi вершини p i q сполученi ребром, якщо G мiстить елемент порядку pq.
Доведено, що у випадку, коли \Gamma (G) = \Gamma (G2(5)), група G мiстить нормальну пiдгрупу N таку, що \pi (N) \subseteq \{ 2, 3, 5\}
та G/N \sim = G2(5).
1. Introduction. For n \in N, let \pi (n) denote the set of all the prime divisors of n, and for a finite
group G let us set \pi (G) = \pi (| G| ). The prime graph \Gamma (G) of a finite group G is a simple graph
with vertex set \pi (G) in which two distinct vertices p and q are joined by an edge if and only if
G has an element of order pq. It is clear that a knowledge of w(G) determines \Gamma (G) completely
but not vise-versa in general. Given a finite group G, the number of nonisomorphic classes of finite
groups H with \Gamma (G) = \Gamma (H) is denoted by h\Gamma (G). If h\Gamma (G) = 1, then G is said to be recognizable
by prime graph. If h\Gamma (G) = k < \infty , then G is called k-recognizable by prime graph, in case
h\Gamma (G) = \infty the group G is called nonrecognizable by prime graph. Obviously a group recognizable
by spectra need not to be recognizable by prime graph, for example A5 is recognizable by spectra
but \Gamma (A5) = \Gamma (A6).
The number of connected components of \Gamma (G) is denoted by s(G). As a consequence of the
classification of the finite simple groups it is proved in [19] and [9], that for any finite simple group
G we have s(G) \leq 6. Let \pi i = \pi i(G), 1 \leq i \leq s, be the connected components of G. For a group
of even order we let 2 \in \pi 1. Recognizability of groups by prime graph was first studied in [5] where
some sporadic simple groups were characterized by prime graph. As another concept we say that a
non-Abelian simple group G is quasirecognizable by graph if every finite group whose prime graph
is \Gamma (G) has a unique non-Abelian composition factor isomorphic to G.
It is proved in [20] that the simple groups G2(7) and 2G2(q), q = 32m+1 > 3, are recognizable
by prime graph, where both groups have disconnected prime graphs. A series of interesting results
concerning recognition of finite simple groups were obtained by B. Khosravi et al. In particular they
have stablished quasirecognizability of the group L10(2) by graph and the recognizability of L16(2)
by graph in [7] and [8], where both groups have connected prime graphs.
Next we introduce useful notation. Let p be a prime number. The set of all non-Abelian finite
simple groups G such that p \in \pi (G) \subseteq \{ 2, 3, 5, . . . , p\} is denoted by Sp. It is clear that the set of
all non-Abelian finite simple groups is the disjoint union of the finite sets Sp for all primes p. The
sets Sp, where p is a prime less than 1000 is given in [21].
2. Preliminary results. Let G be a finite group with disconnected prime graph. The structure
of G is given in [19] which is stated as a lemma here.
c\bigcirc P. NOSRATPOUR, M. R. DARAFSHEH, 2016
1142 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
CHARACTERIZATION OF THE GROUP G2(5) BY THE PRIME GRAPH 1143
Lemma 2.1. Let G be a finite group with disconnected prime graph. Then G satisfies one of
the following conditions:
(a) s(G) = 2, G = KC is a Frobenius group with kernel K and complement C, and the two
connected components of G are \Gamma (K) and \Gamma (C). Moreover K is nilpotent, and here \Gamma (K) is a
complete graph.
(b) s(G) = 2 and G is a 2-Frobeuius group, i.e., G = ABC, where A,AB\unlhd G, B\unlhd BC, and
AB, BC are Frobenius groups.
(c) There exists a non-Abelian simple group P such that P \leq G = G/N \leq \mathrm{A}\mathrm{u}\mathrm{t}(P ) for
some nilpotent normal \pi 1(G)-subgroup N of G and G/P is a \pi 1(G)-group. Moreover, \Gamma (P ) is
disconnected and s(P ) \geq s(G).
If a group G satisfies condition (c) of the above lemma we may write P = B/N, B \leq G, and
G/P = G/B = A, hence in terms of group extensions G = N \cdot P \cdot A, where N is a nilpotent
normal \pi 1(G)-subgroup of G and A is a \pi 1(G)-group.
The above structure lemma was extended to groups with connected prime graphs satisfying certain
conditions [17]. Denote by t(G) the maximal number of primes in \pi (G) pairwise nonadjacent in
\Gamma (G).
In the following we list some properties of the Frobenius group where some of its proof can be
found in [15].
Lemma 2.2. Let G be a Frobenius group with kernel K and complement H. Then:
(a) K is nilpotent and | H| | (| K| - 1).
(b) The connected components of G are \Gamma (K) and \Gamma (H).
(c) | \mu (K)| = 1 and \Gamma (K) is a complete graph.
(d) If | H| is even, then K is Abelian.
(e) Every subgroup of H of order pq, p and q not necessary distinct primes, is cyclic. In
particular if H is Abelian, then it would be cyclic.
(f) If H is nonsolvable, then there is a normal subgroup H0 of H such that [H : H0] \leq 2 and
H0
\sim = SL2(5)\times Z, where every Sylow subgroup of Z is cyclic and | Z| is prime to 2, 3 and 5.
A Frobenius group with cyclic kernel of order m and cyclic complement of order n is denoted
by m : n.
The following result is also used in this paper whose proof is included in [3].
Lemma 2.3. Every 2-Frobenius group is solvable.
Lemma 2.4 [6]. Let G be a finite solvable group all of whose elements are of prime power
order, then the order of G is divisible by at most two distinct primes.
Lemma 2.5 [12]. Let G be a finite group, K \unlhd G, and let G/K be a Frobenius group with
kernel F and cyclic complement C. If (| F | , | K| ) = 1 and F dose not lie in (K \cdot CG(K))/K, then
r \cdot | C| \in w(G) for some prime divisor r of | K| .
Lemma 2.6 [18]. Ln(q) contains a Frobenius subgroup with kernel of order qn - 1 and cyclic
complement of order (qn - 1 - 1)/(n, q - 1).
Using [1], we can find \mu (G2(5)) = \{ 20, 21, 24, 25, 30, 31\} . Therefore, the prime graph of G2(5)
is as a follows.
Our main results are the followin theorem.
Theorem 2.1. If G is a finite group such that \Gamma (G) = \Gamma (G2(5)), then G has a normal subgroup
N such that \pi (N) \subseteq \{ 2, 3, 5\} and G/N \sim = G2(5).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1144 P. NOSRATPOUR, M. R. DARAFSHEH
\bullet \bullet
\bullet
\bullet
\bullet 31
3
2
7
5
Fig. 1. The prime graph of G2(5).
3. Proof of the theorem. We assume G is a group with \Gamma (G) = \Gamma (G2(5)). By Fig. 1, we have
s(G) = 2, hence, G has disconnected prime graph and we can use Lemma 2.1 here:
(a) G is nonsolvable. If G is solvable, then consider a \{ 5, 7, 31\} -Hall subgroup of G and call it
H. By Fig. 1, H dose not contain elements of order 5 \cdot 7, 7 \cdot 31, 5 \cdot 31, and since it is solvable, by
Lemma 2.4 we deduce | t(H)| \leq 2, a contradiction.
(b) G is neither a Frobenius nor a 2-Frobenius group. By (a) and Lemma 2.3, G is not a
2-Frobenius group. If G is a Frobenius group, then by Lemma 2.1, G = KC with Frobenius
kernel K and Frobenius complement C with connected components \Gamma (K) and \Gamma (C). Obviously
\Gamma (K) is a graph with vertex \{ 31\} and \Gamma (C) with vertex set \{ 2, 3, 5, 7\} . Since G is nonsolvable, by
Lemma 2.2(a) C must be nonsolvable. Therefore, by Lemma 2.2(f) C has a subgroup isomorphic
to H0 and [C : H0] \leq 2, where H0
\sim = SL2(5) \times Z with Z cyclic of order prime to 2, 3, 5. But
\mu (SL2(5)) = \{ 4, 6, 10\} from which we can observe that H0 has no element of order 15. This implies
that C has no element of order 15, contradicting Fig. 1.
Conditions (a) and (b) imply that case (c) of Lemma 2.1 holds for G. Hence, there is a non-
Abelian simple group P such that P \leq G = G/N \leq \mathrm{A}\mathrm{u}\mathrm{t}(P ) where N is a nilpotent normal
\pi 1(G)-subgroup of G and G/P is a \pi 1(G)-group and s(P ) \geq 2. We have \pi 1(G) = \{ 2, 3, 5, 7\} and
\pi (G) = \{ 2, 3, 5, 7, 31\} . Therefore, P is a simple group with \pi (P ) \subseteq \{ 2, 3, 5, 7, 31\} , i.e., P \in Sp
where p is a prime number satisfying p \leq 31, p \not = 11, 13, 17, 19, 23, 29. Using [21] we list the
possibilities for P in Table 1.
(c) \{ 31\} \subseteq \pi (P ). By Table 1, | \mathrm{O}\mathrm{u}\mathrm{t}(P )| is a number of the form 2\alpha \cdot 3\beta , therefore, if G/N = P \cdot S,
where S \leq \mathrm{O}\mathrm{u}\mathrm{t}(P ), then | P | p = | G/N | p/| S| p for all p \in \pi (G), where np denotes the p-part of the
integer n \in N. Hence, | N | p =
| G| p
| P | p| S| p
, from which the claim follows because \pi (N) \subseteq \{ 2, 3, 5, 7\} .
Therefore only the following possibilities arise for P : L2(31), L5(2), L6(2), L3(5), L2(5
3) and
G2(5).
(d) P \sim = G2(5). By [4], we have \mu (L5(2)) = \{ 8, 12, 14, 15, 21, 31\} and \mu (L6(2)) = \{ 8, 12, 28,
30, 31, 63\} . Therefore, if P \sim = L5(2) or L6(2), then we have 2 \sim 7 in \Gamma (G), is a contradiction.
By [10], we have \mu (L2(5
3)) = \{ 5, 62, 63\} . Therefore, if P \sim = L2(5
3), then we have 2 \sim 31 in
\Gamma (G), a contradiction.
By [1], we have \mu (L2(31)) = \{ 15, 16, 31\} . Therefore, if P \sim = L2(31), then 7 \in \pi (N). By
Lemma 2.6, P has a Frobenius subgroup 31 : 15, then, by Lemma 2.5, G has an element of order
5 \cdot 7, a contradiction.
By [1], we have \mu (L3(5)) = \{ 20, 24, 31\} . Therefore, if P \sim = L3(5), then 7 \in \pi (N). By
Lemma 2.6, P has a Frobenius subgroup 25 : 24, then, by Lemma 2.5, G has an element of order
2 \cdot 7, a contradiction. Therefore P \sim = G2(5).
(e) G/N \sim = G2(5). So far we proved that P \leq G/N \leq \mathrm{A}\mathrm{u}\mathrm{t}(P ) where P \sim = G2(5). But
\mathrm{A}\mathrm{u}\mathrm{t}(G2(5)) = G2(5), therefore, G/N \sim = G2(5).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
CHARACTERIZATION OF THE GROUP G2(5) BY THE PRIME GRAPH 1145
Table 1. Simple groups in Sp, p \leq 31, p \not = 11, 13, 17, 19, 23, 29
P | P | | \mathrm{O}\mathrm{u}\mathrm{t}(P )|
A5 22 \cdot 3 \cdot 5 2
A6 23 \cdot 32 \cdot 5 4
S4(3) 26 \cdot 34 \cdot 5 2
L2(7) 23 \cdot 3 \cdot 7 2
L2(8) 23 \cdot 32 \cdot 7 3
U3(3) 25 \cdot 33 \cdot 7 2
A7 23 \cdot 32 \cdot 5 \cdot 7 2
L2(49) 24 \cdot 3 \cdot 52 \cdot 72 4
U3(5) 24 \cdot 32 \cdot 53 \cdot 7 6
L3(4) 26 \cdot 32 \cdot 5 \cdot 7 12
A8 26 \cdot 32 \cdot 5 \cdot 7 2
A9 26 \cdot 34 \cdot 5 \cdot 7 2
P | P | | \mathrm{O}\mathrm{u}\mathrm{t}(P )|
J2 27 \cdot 33 \cdot 52 \cdot 7 2
A10 27 \cdot 34 \cdot 52 \cdot 7 2
U4(3) 27 \cdot 36 \cdot 5 \cdot 7 8
S4(7) 28 \cdot 32 \cdot 52 \cdot 74 2
S6(2) 29 \cdot 34 \cdot 5 \cdot 7 1
O+
8 (2) 212 \cdot 35 \cdot 52 \cdot 7 6
L2(31) 25 \cdot 3 \cdot 5 \cdot 31 2
L3(5) 25 \cdot 3 \cdot 53 \cdot 31 2
L2(5
3) 22 \cdot 32 \cdot 53 \cdot 7 \cdot 31 6
G2(5) 26 \cdot 33 \cdot 56 \cdot 7 \cdot 31 1
L5(2) 210 \cdot 32 \cdot 5 \cdot 7 \cdot 31 2
L6(2) 215 \cdot 34 \cdot 5 \cdot 72 \cdot 31 2
(f) \pi (N) \subseteq \{ 2, 3, 5\} . We know that N is a nilpotent normal \{ 2, 3, 5, 7\} -subgroup of G. Regarding
Fig. 1 we obtain:
if 2, 5 | | N | , then \pi (N) \subseteq \{ 2, 3, 5\} ;
if 3 | | N | , then \pi (N) \subseteq \{ 2, 3, 5, 7\} ;
if 7 | | N | , then \pi (N) \subseteq \{ 3, 7\} .
Now we observe that the group G2(5) contains Frobenius subgroup 31 : 5. We may assume N
is elementary Abelian p-group for p \in \{ 2, 3, 5, 7\} . Now if 7 | | N | , then by Lemma 2.5, G has an
element of order 5 \cdot 7, a contradiction. Therefore, \pi (N) \subseteq \{ 2, 3, 5\} .
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Received 24.03.13,
after revision — 21.06.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
|
| id | umjimathkievua-article-1909 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:01Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a0/d36bc885e0362612c22c72af6d7116a0.pdf |
| spelling | umjimathkievua-article-19092019-12-05T09:31:14Z Characterization of the group $G_2(5)$ by the prime graph Характеризацiя групи $G_2(5)$ за допомогою простого графа Darafsheh, M. R. Nosratpour, P. Дарафшех, М. Р. Носратпур, П. Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma (G)$ with vertex set $\pi (G)$ and the set of all prime divisors of $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $G\Gamma (G) = \Gamma (G_2(5))$, then $G$ has a normal subgroup $N$ such that $\pi (N) \subseteq \{ 2, 3, 5\}$ and $G/N \sim = G_2(5)$. Нехай $G$ — скiнченна група. Простим графом $G$ називається граф $\Gamma (G)$ з множиною вершин $\pi (G)$ та множиною всiх простих дiльникiв $|G|$, в якому двi рiзнi вершини $p$ i $q$ сполученi ребром, якщо $G$ мiстить елемент порядку $pq$. Доведено, що у випадку, коли $\Gamma (G) = \Gamma (G_2(5))$, група $G$ мiстить нормальну пiдгрупу $N$ таку, що $\pi (N) \subseteq \{ 2, 3, 5\}$ та $G/N \sim = G_2(5)$. Institute of Mathematics, NAS of Ukraine 2016-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1909 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 8 (2016); 1142-1146 Український математичний журнал; Том 68 № 8 (2016); 1142-1146 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1909/891 Copyright (c) 2016 Darafsheh M. R.; Nosratpour P. |
| spellingShingle | Darafsheh, M. R. Nosratpour, P. Дарафшех, М. Р. Носратпур, П. Characterization of the group $G_2(5)$ by the prime graph |
| title | Characterization of the group $G_2(5)$ by the prime graph |
| title_alt | Характеризацiя групи $G_2(5)$ за допомогою простого графа |
| title_full | Characterization of the group $G_2(5)$ by the prime graph |
| title_fullStr | Characterization of the group $G_2(5)$ by the prime graph |
| title_full_unstemmed | Characterization of the group $G_2(5)$ by the prime graph |
| title_short | Characterization of the group $G_2(5)$ by the prime graph |
| title_sort | characterization of the group $g_2(5)$ by the prime graph |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1909 |
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