Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph
Let $G$ be a finite group. The prime graph of $G$ is the graph $\Gamma(G)$ whose vertex set is the set $\Pi(G)$ of all prime divisors of the order $|G|$ and two distinct vertices $p$ and $q$ of which are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $S$ denotes one of t...
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2012
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508483597631488 |
|---|---|
| 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. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:29:46Z |
| description | Let $G$ be a finite group. The prime graph of $G$ is the graph $\Gamma(G)$ whose vertex set is the set $\Pi(G)$ of all prime divisors of the order $|G|$
and two distinct vertices $p$ and $q$ of which are adjacent by an edge if $G$ has an element of order $pq$.
We prove that if $S$ denotes one of the simple groups $L_5(4)$ and $U_4(4)$ and if $G$ is a finite group with $\Gamma(G) = \Gamma(S)$, then $G$ has a
$G$ normal subgroup $N$ such that $\Pi(N) \subseteq \{2, 3, 5\}$ and $\cfrac GN \cong S$. |
| first_indexed | 2026-03-24T02:25:56Z |
| format | Article |
| fulltext |
UDC 512.5
P. Nosratpour (Islamic Azad Univ. Tehran, Iran),
M. R. Darafsheh (School Math., Statistics and Comput. Sci., College Sci., Univ. Tehran, Iran)
RECOGNITION OF THE GROUPS L5(4) AND U4(4) BY THE PRIME GRAPH
РОЗПIЗНАВАННЯ ГРУП L5(4) ТА U4(4) ПО ГРАФУ ПРОСТИХ ЧИСЕЛ
Let G be a finite group. The prime graph of G is the graph Γ(G) whose vertex set is the set Π(G) of all prime divisors of
the order |G| and two distinct vertices p and q of which are adjacent by an edge if G has an element of order pq. We prove
that if S denotes one of the simple groups L5(4) and U4(4) and if G is a finite group with Γ(G) = Γ(S), then G has a
normal subgroup N such that Π(N) ⊆ {2, 3, 5} and
G
N
∼= S.
Нехай G — скiнченна група. Графом простих чисел групи G називають граф Γ(G), множиною вершин якого є
множина Π(G) усiх простих дiльникiв порядку |G| i в якому двi рiзнi вершини p та q з’єднанi ребром, якщо G
мiстить елемент порядку pq. Доведено, що, якщо S є однiєю з простих груп L5(4) та U4(4), а G є скiнченною
групою, для якої Γ(G) = Γ(S), то G має нормальну пiдгрупу N таку, що Π(N) ⊆ {2, 3, 5} та
G
N
∼= S.
1. Introduction. Let G be a finite group. The spectrum ω(G) of G is the set of orders of elements
in G, where each possible order element occurs once in ω(G) regardless of how many elements
of that order G has. This set is closed and partially ordered by divisibility, hence it is uniquely
determined by its maximal elements. The set of maximal elements of ω(G) is denoted by µ(G). The
number of isomorphic classes of finite groups H such that ω(G) = ω(H) is denoted by h(G). If
h(G) = k ≥ 1 is finite then the group G is called a k-recognizable group by spectrum. If h(G) is not
finite, G is called non-recognizable. A 1-recognizable group is usually called a recognizable group.
The recognizability of finite groups by spectrum was first considered by W. J. Shi et al. in [16]. A list
of finite simple groups which are known to be or not to be recognizable by spectrum is given in [11].
For n ∈ N, let Π(n) denote the set of all the prime divisors of n, and for a finite group G let
us set Π(G) = Π(|G|). The prime graph Γ(G) of a finite group G is a simple graph with vertex set
Π(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 ω(G) determines Γ(G) completely but not vise-versa
in general. Given a finite group G, the number of non-isomorphic classes of finite groups H with
Γ(G) = Γ(H) is denoted by hΓ(G). If hΓ(G) = 1, then G is said to be recognizable by prime graph.
If hΓ(G) = k <∞, then G is called k-recognizable by prime graph, in case hΓ(G) =∞ the group
G is called non-recognizable by graph. Obviously a group recognizable by spectra need not to be
recognizable by prime graph, for example A5 is recognizable by spectra but Γ(A5) = Γ(A6).
The number of connected components of Γ(G) is denoted by s(G). As a consequence of the
classification of the finite simple groups it is proved in [19] and [10], that for any finite simple group
G we have s(G) ≤ 6. Let Πi = Πi(G), 1 ≤ i ≤ s, be the connected components of G. For a group
of even order we let 2 ∈ Π1. Recognizability of groups by prime graph was first studied in [6] where
some sporadic simple groups were characterized by prime graph. As another concept we say that a
non-abelian simple group G is quasi-recognizable by graph if every finite group whose prime graph
is Γ(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 graph, where both groups have disconnected prime graphs. A series of interesting results concern-
c© P. NOSRATPOUR, M. R. DARAFSHEH, 2012
210 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
RECOGNITION OF THE GROUPS L5(4) AND U4(4) BY THE PRIME GRAPH 211
ing recognition of finite simple groups were obtained by B.Khosravi et al. In particular they have
stabilized quasi-recognizability of the group L10(2) by graph and the recognizability of L16(2) by
graph in [8] and [9], 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 ∈ Π(G) ⊆ {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.
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 Γ(K) and Γ(C). Moreover K is nilpotent, and here Γ(K) is a
complete graph.
b) s(G) = 2 and G is a 2-Frobeuius group, i.e., G = ABC where A,AB � G, B � BC, and
AB, BC are Frobenius groups.
c) There exists a non-abelian simple group P such that P ≤ G =
G
N
≤ Aut (P ) for some
nilpotent normal Π1(G)-subgroup N of G and
G
P
is a Π1(G)-group. Moreover, Γ(P ) is disconnected
and s(P ) ≥ s(G).
If a group G satisfies condition(c) of the above lemma we may write P =
B
N
, B ≤ G, and
G
P
=
G
B
= A, hence in terms of group extensions G = N · P · A, where N is a nilpotent normal
Π1(G)-subgroup of G and A is a Π1(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 Π(G) pairwise nonadjacent
in Γ(G) and t(2, G) the maximal number of primes in Π(G) nonadjacent to 2.
Lemma 2.2. Let G be a finite group satisfying the following conditions:
a) there exist three pairwise distinct primes in Π(G) nonadjacent in Γ(G), i.e., t(G) ≥ 3.
b) there exists an odd prime in Π(G) nonadjacent in Γ(G) to 2, 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 L2(q) for some odd q, and t(S) = t(2, G) = 3.
2. For every prime p ∈ Π(G) nonadjacent to 2 in Γ(G) a Sylow p-subgroups of G is isomorphic
to a Sylow p-subgroup of S. In particular t(2, S) ≥ t(2, G).
In the following we list some properties of the Frobenius group where some of its proof can be
found in [15].
Lemma 2.3. 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 Γ(K) and Γ(H).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
212 P. NOSRATPOUR, M. R. DARAFSHEH
c) |µ(K)| = 1 and Γ(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 non-solvable, then there is a normal subgroup H0 of H such that [H : H0] ≤ 2 and
H0
∼= SL2(5)× 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 [4].
Lemma 2.4. Every 2-Frobenius group is solvable.
Lemma 2.5 [7]. 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 2.6 [12]. Let G be a finite group, K �G, and let
G
K
be a Frobenius group with kernel
F and cyclic complement C. If (|F |, |K|) = 1 and F does not lie in
K · CG(K)
K
, then r · |C| ∈ w(G)
for some prime divisor r of |K|.
Lemma 2.7 [18]. (1) If there exists a primitive prime divisor r of qn − 1, then Ln(q) has a
Frobenius subgroup with kernel of order r and cyclic complement of order n.
(2) Ln(q) contains a Frobenius subgroup with kernel of order qn−1 and cyclic complement of
order
qn−1 − 1
(n, q − 1)
.
Using [3] we can find µ(L5(4)) and using [13] we can find µ(U4(4)).
Lemma 2.8. For the groups L5(4) and U4(4) we have
µ(L5(4)) = {8, 60, 126, 255, 315, 341},
µ(U4(4)) = {51, 65, 30, 20}.
Using Lemma 2.8 we can draw the prime graphs of the groups L5(4) and U4(4) (see Figures 1
and 2).
Our main results are the following:
Theorem 2.1. If G is a finite group such that Γ(G) = Γ(L5(4)), then G has a normal subgroup
N such that Π(N) ⊆ {2, 3, 5} and
G
N
∼= L5(4).
Theorem 2.2. If G is a finite group such that Γ(G) = Γ(U4(4)), then G has a normal subgroup
N such that Π(N) ⊆ {2, 3, 5} and
G
N
∼= U4(4).
3. Proof of Theorem 2.1. First we prove Theorem 2.1 in series of steps. Therefore we assume
G is a group with Γ(G) = Γ(PSL5(4)). By Fig. 1 we have s(G) = 2, hence G has disconnected
prime graph and we can use the structure theorem for G which is denoted by Lemma 2.1 here:
a) G is non-solvable.
If G is solvable, then consider a {7, 11, 17}-Hall subgroup of G and call it H. By Fig. 1, H
dose not contain elements of order 7 · 11, 7 · 17, 11 · 17, and since it is solvable, by [7] we deduce
Π(H) ≤ 2, a contradiction.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
RECOGNITION OF THE GROUPS L5(4) AND U4(4) BY THE PRIME GRAPH 213
•
•
•
• •
• •7
5
17
3
2 11 31
Fig. 1. The prime graph of L5(4).
•
•
•
• •
13
5
2
3
17
Fig. 2. The prime graph of U4(4).
b) G is neither a Frobenius nor a 2-Frobenius group.
By (a) and Lemma 2.4, 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 Γ(K) and Γ(C). Obviously Γ(K) is a graph with vertices {11, 31} and Γ(C) with vertex
set {2, 3, 5, 7, 17}. Since G is non-solvable, by Lemma 2.3(a) C must be non-solvable. Therefore, by
Lemma 2.3(f) C has a subgroup isomorphic to H0 and [C : H0] ≤ 2, where H0
∼= SL2(5)× Z with
Z cyclic of order prime to 2, 3, 5. But µ(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.
(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 ≤ G =
G
N
≤ Aut (P ) where N is a nilpotent normal Π1(G)-
subgroup of G and
G
P
is a Π1(G)-group and s(P ) ≥ 2. We have Π1(G) = {2, 3, 5, 7, 17} and
Π(G) = {2, 3, 5, 7, 11, 17, 31}. Therefore P is a simple group with Π(P ) ⊆ {2, 3, 5, 7, 11, 17, 31},
i.e., P ∈ Sp where p is a prime number satisfying p ≤ 31, p 6= 13, 19, 23, 29. Using [21] we list the
possibilities for P in Table 1.
c) {11, 31} ⊆ Π(P ).
By Table 1, |Out (P )| is a number of the form 2α · 3β · 5γ , therefore if
G
N
= P · S where
S ≤ Out (P ), then |P |p =
∣∣∣∣GN
∣∣∣∣
p
/|S|p for all p ∈ Π(G), where np denotes the p-part of the integer
n ∈ N. Hence |N |p =
|G|p
|P |p.|S|p
, from which the claim follows because Π(N) ⊆ {2, 3, 5, 7, 17}.
Therefore, only the following possibilities arise for P : L2(32), L5(4), O+
12(2), S10(2).
d) P ∼= L5(4).
By [14] the group L2(32) has three prime graph components as follows Π1 = {2}, Π2 = {31}
and Π3 = {3, 11}. Both groups S10(2) and O+
12(2) have two prime graph components with the
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
214 P. NOSRATPOUR, M. R. DARAFSHEH
Table 1. Simple groups in Sp, p ≤ 31, p 6= 13, 19, 23, 29.
P |P | |out (P )|
A5 22 · 3 · 5 2
A6 23 · 32 · 5 4
S4(3) 26 · 34 · 5 2
L2(7) 23 · 3 · 7 2
L2(8) 23 · 32 · 7 3
U3(3) 25 · 33 · 7 2
A7 23 · 32 · 5 · 7 2
L2(49) 24 · 3 · 52 · 72 4
U3(5) 24 · 32 · 53 · 7 6
L3(4) 26 · 32 · 5 · 7 12
A8 26 · 32 · 5 · 7 2
A9 26 · 34 · 5 · 7 2
J2 27 · 33 · 52 · 7 2
A10 27 · 34 · 52 · 7 2
U4(3) 27 · 36 · 5 · 7 8
S4(7) 28 · 32 · 52 · 74 2
S6(2) 29 · 34 · 5 · 7 1
O+
8 (2) 212 · 35 · 52 · 7 6
L2(11) 22 · 3 · 5 · 11 2
M11 24 · 32 · 5 · 11 1
M12 26 · 33 · 5 · 11 2
U5(2) 210 · 35 · 5 · 11 2
M22 27 · 32 · 5 · 7 · 11 2
A11 27 · 34 · 52 · 7 · 11 2
M cL 26 · 36 · 53 · 7 · 11 2
P |P | |out (P )|
HS 29 · 32 · 53 · 7 · 11 2
A12 29 · 35 · 52 · 7 · 11 2
U6(2) 215 · 36 · 5 · 7 · 11 6
L2(17) 24 · 32 · 17 2
L2(16) 24 · 3 · 5 · 17 4
S4(4) 28 · 32 · 52 · 17 4
He 210 · 33 · 52 · 73 · 17 2
O−8 (2) 212 · 34 · 5 · 7 · 17 2
L4(4) 212 · 34 · 52 · 7 · 17 4
S8(2) 216 · 35 · 52 · 7 · 17 1
O−10(2) 220 · 36 · 52 · 7 · 11 · 17 2
L2(31) 25 · 3 · 5 · 31 2
L3(5) 25 · 3 · 53 · 31 2
L2(32) 25 · 3 · 11 · 31 5
L2(53) 22 · 32 · 53 · 7 · 31 6
G2(5) 26 · 33 · 56 · 7 · 31 1
L5(2) 210 · 32 · 5 · 7 · 31 2
L6(2) 215 · 34 · 5 · 72 · 31 2
O+
10(2) 220 · 35 · 52 · 7 · 17 · 31 2
L5(4) 220 · 35 · 52 · 7 · 11 · 17
31 4
S10(2) 225 · 36 · 52 · 7 · 11 · 17
31 1
O+
12(2) 230 · 38 · 52 · 72 · 11 · 17
31 2
second component Π2 = {31}. In any case the above facts violates the prime graph of L5(4) in
Fig. 1, and this completes our claim.
e)
G
N
∼= L5(4). So far we proved that P ≤ G
N
≤ Aut (P ) where P ∼= L5(4). But Aut (L5(4)) =
= L5(4) : A where A is a four group. If σ2 denotes the field automorphism and Θ the graph auto-
morphism of L5(4), then A = 〈σ2,Θ〉 and we have the following possibilities for
G
N
:
G
N
∼= L5(4),
G
N
∼= L5(4) : 〈σ2〉,
G
N
∼= L5(4) : 〈Θ〉,
G
N
∼= L5(4) : 〈σ2 ·Θ〉 or
G
N
∼= L5(4) : 〈σ2,Θ〉.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
RECOGNITION OF THE GROUPS L5(4) AND U4(4) BY THE PRIME GRAPH 215
It is shown in [5] that all the above possibilities except
G
N
∼= L5(4) violates the structure of the
prime graph of G in Fig. 1, therefore our claim is proved.
f) Π(N) ⊆ {2, 3, 5}.
We know that N is a nilpotent normal {2, 3, 5, 7, 17}-subgroup of G. Regarding Fig. 1 we obtain:
If 2 | |N |, then Π(N) ⊆ {2, 3, 5, 7}.
If 17 | |N |, then Π(N) ⊆ {3, 5, 17}.
If 7 | |N |, then Π(N) ⊆ {2, 3, 5, 7}.
If 7 | |N | we may assume M is the characteristic 7′-subgroup of N such that
H
K
∼= L5(4), where
H =
G
M
and K =
N
M
is a non-trivial 7-group. By Lemma 2.7(1) L5(4) has a Frobenius group of
the shape 44 : 255, where 44 denotes Z4
4 and is the Frobenius kernel and 255 is the cyclic group of
order 5 · 3 · 17 and is the Frobenius complement. Now by Lemma 2.6, H would have an element of
order 7 · 17 violating Fig. 1. Also L5(4) has a Frobenius group of the shape 11: 2, then, if 17 | |N |.
Therefore by Lemma 2.6, H would have an element of order 2 · 17 violating Fig. 1. Therefore, the
only possibility is Π(N) ⊆ {2, 3, 5}.
Theorem 2.1 is proved.
Proof of Theorem 2.2. Therefore we will assume that G is a group with Γ(G) = Γ(U4(4)). By
Fig. 2 we have s(G) = 1, i.e. the prime graph of G is connected. In this case Lemma 2.2 is applicable
for the structure of G, because {2, 13, 17} is an independent set as well as a 2-independent set for
G, hence t(G) = 3 and t(2, G) = 3. Therefore 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.
Before we continue our investigation, we need a table similar to Table 1 for simple groups G
with 13 ∈ Π(G) ⊆ {2, 3, 5, . . . , 13} but 7 - |G|, 11 - |G|. Using [21] we obtain Table 2.
Now suppose G satisfies condition (a) of Lemma 2.2. We have S � A7 because 7 - |G|. If
S ∼= L2(q), q odd, then by Tables 1 and 2 we obtain S ∼= L2(5), L2(9), L2(17) or L2(25). Regarding
the order of outer automorphism of the groups S listed above we obtain the following facts:
If S ∼= PSL2(5) or PSL2(9), then {13, 17} ⊆ Π(K).
If S ∼= PSL2(17), then {13} ⊆ Π(K).
If S ∼= PSL2(25), then {17} ⊆ Π(K).
Table 2. Simple groups G with 13 ∈ Π(G) ⊆ {2, 3, . . . , 13, 17} but 7, 11 - |G|.
S |S| |out (S)|
A5
∼= L2(5) 22 · 3 · 5 2
A6
∼= L2(9) 23 · 32 · 5 4
L3(3) 24 · 33 · 13 2
L2(25) 23 · 3 · 52 · 13 4
U3(4) 26 · 3 · 52 · 13 4
S |S| |out (S)|
S4(5) 26 · 32 · 54 · 13 2
L4(3) 27 · 36 · 5 · 13 4
2F4(2)′ 211 · 33 · 52 · 13 2
U4(4) 212 · 32 · 53 · 13 · 17 4
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
216 P. NOSRATPOUR, M. R. DARAFSHEH
Now by Lemma 2.7(2), PSL2(q) has a Frobenius group of the shape q :
q − 1
2
. Since
q − 1
2
for
q = 5, 9, 17, 25 is even, Lemma 2.6 implies that G has an element of order 2 · 13 or 2 · 17, both
contradicting Fig. 2.
Therefore, G must satisfy condition (b) of Lemma 2.2. The primes non-adjacent to 2 are 13 and
17, hence {13, 17} ⊆ Π(S), and regarding Tables 1 and 2 the only simple group whose order is
divisible by 13 and 17 is U4(4). Therefore we obtain U4(4) ≤ G
K
≤ Aut (U4(4)).
Now we observe that the group U4(4) contains Frobenius subgroups of types 17: 4 and 13: 3.
We may assume K is elementary abelian p-group for p ∈ {2, 3, 5, 13, 17}. Therefore by Lemma 2.6
and Fig. 2 the orders of K can not be divisible by 13. By Lemma 2.7 in [14] we have 17 - |K|.
Therefore Π(K) ⊆ {2, 3, 5}.
By [2] the outer automorphism group of U4(4) is a cyclic group isomorphic to Z4, hence we
have the following lemma:
Lemma 4.1. If G is an almost simple group related to L = U4(4), then G is isomorphic to
one of the following groups: L, L : 2 or L : 4.
If U4(4) ≤ G
K
≤ U4(4) : 4, then by above lemma, we have
G
K
= U4(4), U4(4) : 2 or U4(4) : 4.
If
G
K
= U4(4) : 2, then let t denote the outer automorphism of order 2, by [1] we have C(t)
U4(4) =
= S4(4) implying that t centralizes an element of order 17 violating Fig. 2.
If
G
K
= U4(4) : 4, then, similar to the above case, let t denote the outer automorphism of order
4, by [1] we have C(t)
U4(4) = S4(4) implying that t centralizes an element of order 17 violating Fig. 2.
Therefore, the only possibility is
G
K
∼= U4(4).
Theorem 2.2 is proved.
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| id | umjimathkievua-article-2567 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:56Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a3/8d30fe3f4a59b43d1e2064fd1d1a77a3.pdf |
| spelling | umjimathkievua-article-25672020-03-18T19:29:46Z Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph Розпiзнавання груп $L_5(4)$ та $U_4(4)$ по графу простих чисел Darafsheh, M. R. Nosratpour, P. Дарафшех, М. Р. Носратпур, П. Let $G$ be a finite group. The prime graph of $G$ is the graph $\Gamma(G)$ whose vertex set is the set $\Pi(G)$ of all prime divisors of the order $|G|$ and two distinct vertices $p$ and $q$ of which are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $S$ denotes one of the simple groups $L_5(4)$ and $U_4(4)$ and if $G$ is a finite group with $\Gamma(G) = \Gamma(S)$, then $G$ has a $G$ normal subgroup $N$ such that $\Pi(N) \subseteq \{2, 3, 5\}$ and $\cfrac GN \cong S$. Нехай $G$ — скiнченна група. Графом простих чисел групи $G$ називають граф $\Gamma(G)$, множиною вершин якого є множина $\Pi(G)$ усiх простих дiльникiв порядку $|G|$ i в якому двi рiзнi вершини $p$ та $q$ з’єднанi ребром, якщо $G$ мiстить елемент порядку $pq$. Доведено, що, якщо $S$ є однiєю з простих груп $L_5(4)$ та $U_4(4)$, а $G$ є скiнченною групою, для якої $\Gamma(G) = \Gamma(S)$, то $G$ має нормальну пiдгрупу $N$ таку, що $\Pi(N) \subseteq \{2, 3, 5\}$ та $\cfrac GN \cong S$. Institute of Mathematics, NAS of Ukraine 2012-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2567 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 2 (2012); 210-217 Український математичний журнал; Том 64 № 2 (2012); 210-217 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2567/1893 https://umj.imath.kiev.ua/index.php/umj/article/view/2567/1894 Copyright (c) 2012 Darafsheh M. R.; Nosratpour P. |
| spellingShingle | Darafsheh, M. R. Nosratpour, P. Дарафшех, М. Р. Носратпур, П. Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph |
| title | Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph |
| title_alt | Розпiзнавання груп $L_5(4)$ та $U_4(4)$ по графу простих чисел |
| title_full | Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph |
| title_fullStr | Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph |
| title_full_unstemmed | Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph |
| title_short | Recognition of the groups $L_5(4)$ and $U_4(4)$ by the prime graph |
| title_sort | recognition of the groups $l_5(4)$ and $u_4(4)$ by the prime graph |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2567 |
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