Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$
UDC 512.5 In this paper, it is demonstrated that every finite group $G$ with the same order and degree pattern as $O^{-}_{8}(q)$ for certain $q$ is necessarily isomorphic to the group $O^{-}_{8}(q)$.
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2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508262290423808 |
|---|---|
| author | Bibak, M. Rezaeezadeh, G. H. Bibak, M. Rezaeezadeh, G. H. |
| author_facet | Bibak, M. Rezaeezadeh, G. H. Bibak, M. Rezaeezadeh, G. H. |
| author_sort | Bibak, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-07-15T07:54:29Z |
| description | UDC 512.5
In this paper, it is demonstrated that every finite group $G$ with the same order and degree pattern as $O^{-}_{8}(q)$ for certain $q$ is necessarily isomorphic to the group $O^{-}_{8}(q)$. |
| doi_str_mv | 10.37863/umzh.v74i6.2357 |
| first_indexed | 2026-03-24T02:22:25Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i6.2357
UDC 512.5
M. Bibak (Payame Noor Univ., Tehran, Iran),
G. H. Rezaeezadeh (Univ. Shahrekord, Iran)
CHARACTERIZATION BY ORDER AND DEGREE PATTERN
OF THE SIMPLE GROUPS \bfitO -
\bfeight (\bfitq ) FOR CERTAIN \bfitq
ХАРАКТЕРИСТИКА ПОРЯДКIВ ТА СТЕПЕНIВ
ПРОСТИХ ГРУП \bfitO -
\bfeight (\bfitq ) ДЛЯ ЗАДАНОГО \bfitq
In this paper, it is demonstrated that every finite group G with the same order and degree pattern as O -
8 (q) for certain q
is necessarily isomorphic to the group O -
8 (q).
Доведено, що будь-яка скiнченна група G, яка має тi ж самi порядок та степiнь, що й група O -
8 (q) для деякого q,
необхiдно має збiгатися з O -
8 (q).
1. Introduction. Let G be a finite group, \pi (G) the set of all prime divisors of its order and \pi e(G)
the spectrum of G, that is, the set of its element orders. The Gruenberg – Kegel graph \Gamma (G) or
prime graph of G is a simple graph with vertex set \pi (G), in which two distinct vertices p and q are
adjacent by an edge if and only if pq \in \pi e(G).
For the first time the concept of degree pattern of prime graph was defined in [7]. Let G be
a finite group and \pi (G) = \{ p1, p2, . . . , pk\} with p1 < p2 < . . . < pk. If \mathrm{d}\mathrm{e}\mathrm{g}(p) of a vertex
p \in \pi (G) is the number of edges incident to p, then the degree pattern of G is defined as D(G) =
=
\bigl(
\mathrm{d}\mathrm{e}\mathrm{g}(p1),\mathrm{d}\mathrm{e}\mathrm{g}(p2), . . . ,\mathrm{d}\mathrm{e}\mathrm{g}(pk)
\bigr)
. A finite group G is called k-fold OD-characterizable if there
are exactly k nonisomorphic groups H such that | H| = | G| and D(H) = D(G). Usually a 1-fold
OD-characterizable group is called an OD-characterizable group.
A characterization of the finite group G by degree pattern was defined in [7], in which the
authors proved that all the sporadic simple groups, the alternating groups \BbbA p, where p and p - 2 are
prime numbers, and some simple groups of Lie type are OD-characterizable, however the projective
symplectic group SP6(3) is 2-fold OD-characterizable. In [6, 8, 12], it is shown that some projective
special linear groups are OD-characterizable. In [15], it is proved that the automorphism groups of
orthogonal groups O+
10(2) and O -
10(2) are OD-characterizable. Also, in a series of papers [4, 5, 9, 10],
the characterization by order and degree pattern for some finite almost simple groups has been studied
(recall that a group G is an almost simple group, if S \leq G \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), for some non-Abelian simple
group S ). In this paper, we prove that O -
8 (q) where q \in \{ 3 - 5, 8, 9, 13\} is OD-characterizable.
Throughout this paper, we use the following definition and notions related to \Gamma (G): A set of
vertices of a graph is called independent if its elements are pairwise nonadjacened. We denote by
t(G) the maximal number of vertices in independent sets of \Gamma (G) and by t(r,G) the maximal
number of vertices in independent sets of \Gamma (G) containing a prime r. Denote by s(G) the number of
connected components of \Gamma (G) and by \pi i = \pi i(G), i = 1, 2, . . . , s(G), the ith connected component
of \Gamma (G). If 2 \in \pi (G) we always suppose 2 \in \pi 1.
Also, we use the following notations. For p \in \pi (G), we denote by \mathrm{S}\mathrm{y}\mathrm{l}p(G) and Gp the set
of all Sylow p-subgroups of G and a Sylow p-subgroup of G, respectively. If p is a prime and
c\bigcirc M. BIBAK, G. H. REZAEEZADEH, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6 743
744 M. BIBAK, G. H. REZAEEZADEH
m be a natural number, then we write | m| p for the p-part of m, i.e., the highest power of p that
divides m. Given a prime p, we denote by Sp the set of all finite non-Abelian simple groups G
such that max\pi (G) = p. Note that the full list of all finite non-Abelian simple groups S in Sp for
5 \leq p \leq 997, has been determined in [16]. In this paper, we deal with the finite non-Abelian simple
groups in Sp, where p \in \{ 41, 193, 241, 257, 313, 631\} , and for convenience, we list them in Table 2.
The other unexplained notations are standard and refer to [11].
2. Preliminaries. In this section, we list some basic and known results that will be used.
Definition 2.1. A group G is a 2-Frobenius group if there exists a normal series 1\unlhd H\unlhd K\unlhd G
such that K and G/H are Frobenius groups with kernels H and K/H, respectively.
The structure of finite groups with nonconnected prime graph is described in the following lemma.
Lemma 2.1 (Gruenberg – Kegel theorem of [14]). Let G be a finite group with s(G) \geq 2. Then
one of the following statements holds:
(a) G is a Frobenius or a 2-Frobenius group;
(b) G has a normal series 1 \unlhd H \unlhd K \unlhd G where H is a nilpotent \pi 1-group, K/H is a non-
Abelian simple group and G/K is a \pi 1-group such that | G/K| divides | \mathrm{O}\mathrm{u}\mathrm{t}(K/H)| . Moreover,
each odd-order components of G is also an odd-order component of K/H.
Lemma 2.2 (Corollary 3.8 of [1]). Let G be a finite group with n = | \pi (G)| and let d1 \leq
\leq d2 \leq . . . \leq dn be the degree sequence of \Gamma (G). If d1 + dd1+2 \leq n - 3, then t(G) \geq 3.
Lemma 2.3 (Lemma 2.8 of [6]). Let \Gamma (G) be the prime graph of G with exactly two vertices
of degree 1. Then t(G) \geq 3, if one of the following statements holds:
(1) | \pi (G)| = 6 and \Gamma (G) has at least two vertices of degree 2;
(2) | \pi (G)| \geq 7 and \Gamma (G) has at least two vertices of degree 3.
Lemma 2.4 [13]. Let G be a finite group with t(G) \geq 3, t(2, G) \geq 2, and K be the maximal
normal solvable subgroup of G. Then there exists a non-Abelian simple group S such that S \leq
\leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S).
Lemma 2.5 (Lemma 2.7 of [6]). Let G be a finite group of even order with t(G) \geq 3. Then G
is nonsolvable, and so it is not a 2-Frobenius group. If, moreover, | G| 3 \not = 3 or | G| 5 \not = 5, then G is
not a Frobenius group.
The following two lemmas give a complete description of the spectra of groups O -
2n(q) for all
possible values q.
Lemma 2.6 (Corollaries 8 and 9 of [2]). Let O = O\varepsilon
2n(q), where q be a power of an odd prime
p, n \geq 4 and \varepsilon \in \{ +, - \} . Moreover, assume that d = (4, qn - 1) and c =
d
2
. Then \pi e(O) consists
of all divisors of the following numbers:
(1)
qn - \varepsilon
d
;
(2)
[qn1 - \delta , qn2 - \varepsilon \delta ]
e
, where \delta \in \{ +, - \} , n1, n2 > 0, n1 + n2 = n; e = 2 if (qn1 - \delta )2 =
= (qn2\delta )2 and e = 1 otherwise;
(3) [qn1 - \delta 1, q
n2 - \delta 2, . . . , q
ns - \delta s], where s \geq 3, \delta i \in \{ +, - \} , ni > 0 for all 1 \leq i \leq s,
n1 + . . .+ ns = n and \delta 1\delta 2 . . . \delta s = \varepsilon ;
(4) p
\biggl[
q \pm 1,
qn - 2 + 1
2
\biggr]
;
(5) p[q\pm 1, qn1 - \delta 1, q
n2 - \delta 2, . . . , q
ns - \delta s], where s \geq 2, \delta i \in \{ +, - \} , ni > 0 for all 1 \leq i \leq s,
n1 + . . .+ ns = n - 2 and \delta 1\delta 2 . . . \delta s = \varepsilon ;
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
CHARACTERIZATION BY ORDER AND DEGREE PATTERN OF THE SIMPLE GROUPS O -
8 (q) . . . 745
Table 1. The order and degree pattern of simple groups O -
8 (q) for certain q
S | S| D(S)
O -
8 (3) 210.312.5.7.13.41 (4, 2, 2, 1, 1, 0)
O -
8 (4) 224.34.53.7.13.17.257 (3, 5, 5, 2, 3, 2, 0)
O -
8 (5) 210.34.512.7.13.31.313 (5, 5, 3, 2, 3, 2, 0)
O -
8 (8) 236.37.5.73.13.17.19.73.241 (2, 6, 3, 6, 3, 1, 2, 2, 1)
O -
8 (9) 213.324.53.7.13.17.41.73.193 (6, 3, 6, 3, 3, 1, 3, 2, 1)
O -
8 (43) 210.34.52.73.113.13.17.37.4312.139.193.521.631 (9, 9, 6, 9, 9, 9, 5, 2, 6, 5, 2, 2, 4)
(6) pl
qn1 \pm 1
3
, where l > 0 and pl - 1 + 3 + 2n1 = 2n;
(7) pL
\bigl[
qn1 \pm 1, . . . , qns \pm 1
\bigr]
, where l > 0, s \geq 2 and ni > 0 for all 1 \leq i \leq s and
pl - 1 + 3 + 2(n1 + n2 + . . .+ ns) = 2n;
(8) pl if 2n = pl - 1 + 3 for l > 0.
Lemma 2.7 (Corollary 4 of [2]). Let O = O\varepsilon
2n(q), where q is even, n \geq 4 and \varepsilon \in \{ +, - \} .
The set \pi e(O) consists of all divisors of the following numbers:
(1)
\bigl[
qn1 \pm \tau 1, q
n2 \pm \tau 2, . . . , q
ns \pm \tau s
\bigr]
, where s \geq 1, \tau i \in \{ +, - \} , ni > 0 for all 1 \leq i \leq s,
n1 + . . .+ ns = n and \tau 1\tau 2 . . . \tau s = \varepsilon ;
(4) p
\biggl[
q \pm 1,
qn - 2 + 1
2
\biggr]
;
(5) p
\bigl[
q\pm 1, qn1 - \delta 1, q
n2 - \delta 2, . . . , q
ns - \delta s
\bigr]
, where s \geq 2, \delta \in \{ +, - \} , ni > 0 for all 1 \leq i \leq s,
n1 + . . .+ ns = n - 2 and \delta 1\delta 2 . . . \delta s = e;
(6) pl
qn1 \pm 1
3
, where l > 0 and pl - 1 + 3 + 2n1 = 2n;
(7) pL
\bigl[
qn1 \pm 1, . . . , qns \pm 1
\bigr]
, where l > 0, s \geq 2 and ni > 0 for all 1 \leq i \leq s and
pl - 1 + 3 + 2(n1 + n2 + . . .+ ns) = 2n;
(8) pl if 2n = pl - 1 + 3 for l > 0.
By using Lemmas 2.6, 2.7 and [16], we contain some results which are listed in the Table 1.
Lemma 2.8 (Lemma 2.1 of [12]). Let S be a finite non-Abelian simple group in Sp where 5 \leq
\leq p \leq 997. Then \pi (\mathrm{O}\mathrm{u}\mathrm{t}(S)) \subseteq \{ 2, 3, 5, 7, 11\} .
Lemma 2.9 (Lemma 2.12 of [3]). Let G be a group and N be a normal subgroup of G with
order pn, n \geq 1. If (r, | \mathrm{A}\mathrm{u}\mathrm{t}(N)| ) = 1, where r \in \pi (G), then G has an element of order pr.
3. Main results. In this section, we study the characterization problem for the simple groups
O -
8 (q) with q \in \{ 3, 4, 5, 8, 9, 43\} by their orders and degree patterns.
Proposition 3.1. The orthogonal group O -
8 (3) is OD-characterizable.
Proof. Assume that G be a finite group such that | G| = | O -
8 (3)| = 210.312.5.7.13.41 and
D(G) = D(O -
8 (3)) = (4, 2, 2, 1, 1, 0). By Lemma 2.3, it follows that t(G) \geq 3. Furthermore,
t(2, G) \geq 2 because \mathrm{d}\mathrm{e}\mathrm{g}(2) = 4 and | \pi (G)| = 6. Consequently, from Lemma 2.4 we imply that
there exists a finite non-Abelian simple group S such that S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), where K is the
maximal normal solvable subgroup of G.
We show that K is a \{ 13, 41\} \prime -group. Assume that K is not a \{ 13, 41\} \prime -group. Then either
13 \in \pi (K) or 41 \in \pi (K). Suppose that \{ r, s\} = \{ 13, 41\} , r \in \pi (K) and R is a Sylow r-subgroup
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
746 M. BIBAK, G. H. REZAEEZADEH
of K. Then NG(R) contains an element of order s, so G contains an element of order r.s, which is
a contradiction. Therefore, K is a \{ 13, 41\} \prime -group.
Since S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), it follows that \{ 13, 41\} \subseteq \pi (G/K) \subseteq \pi (\mathrm{A}\mathrm{u}\mathrm{t}(S)). On the other
hand, \pi (\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{S})/S) = \pi (\mathrm{O}\mathrm{u}\mathrm{t}(S)) \cap \{ 13, 41\} = \varnothing by Lemma 2.8. Hence, \{ 13.41\} \subseteq \pi (S) and
so by using the collected results contained in Table 2, we conclude that S is isomorphic to O -
8 (3).
Therefore, O -
8 (3) \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(O -
8 (3)), and since | G| = | O -
8 (3)| , we deduce that | K| = 1 and
G \sim = O -
8 (3).
Proposition 3.2. The orthogonal group O -
8 (4) is OD-characterizable.
Proof. Suppose that G be a finite group such that | G| = | O -
8 (4)| = 224.34.53.7.13.17.257 and
D(G) = D(O -
8 (4)) = (3, 5, 5, 2, 2, 3, 0). Then the prime graph of G has the following form:
13
7
5
3
17
2
257
Fig. 1
Since \{ 13, 17, 257\} is an independent set in \Gamma (G), it follows that t(G) \geq 3. By Lemma 2.5, G
is neither a Frobenius group nor a 2-Frobenius group, and hence Lemma 2.1 implies that G has a
normal series 1\unlhd H \unlhd K \unlhd G, where K/H is a non-Abelian simple group and G/K is a \pi 1-group
such that | G/K|
\bigm| \bigm| | \mathrm{O}\mathrm{u}\mathrm{t}(K/H)| . Moreover, each odd-order components of G is also an odd-order
component of K/H. Thus 257 is an isolated vertex of prime graph of K/H. Now, according to
the results collected in Table 2, we deduce that K/H is isomorphic to one of the following groups:
L2(2
8) or O -
8 (4).
If K/H is isomorphic to L2(2
8), then (| G/K| , 13) = 1 by | \mathrm{O}\mathrm{u}\mathrm{t}(K/H)| = 16 and so the Sylow
13-subgroup of H is of order 13 and is normal in G. Since (257, | \mathrm{A}\mathrm{u}\mathrm{t}(H13)| ) = 1, it follows that
G has an element of order 257.13 by Lemma 2.9, which contradicts our assumption \mathrm{d}\mathrm{e}\mathrm{g}(257) = 0.
Therefore, K/H is isomorphic to O -
8 (4), and since | G| = | O -
8 (4)| , we obtain | H| = 1 and
G \sim = O -
8 (4).
Proposition 3.3. The orthogonal group O -
8 (5) is OD-characterizable.
Proof. Assume that G be a finite group such that | G| = | O -
8 (5)| = 210.34.512.7.13.31.313
and D(G) = D(O -
8 (5)) = (5, 5, 3, 2, 3, 2, 0). According to these conditions on G, we conclude that
\Gamma (G) has the following form:
31
7
2
3
13
5
313
Fig. 2
From the structure of the prime graph of G, as shown in Fig. 2, we deduce that t(G) \geq 3. Hence, by
Lemma 2.5 implies that G is neither a Frobenius group nor a 2-Frobenius group. So, it follows by
Lemma 2.1 that G has a normal series 1\unlhd H \unlhd K \unlhd G, where K/H is a non-Abelian simple group
and G/K is a \pi 1-group such that | G/K|
\bigm| \bigm| \bigm| \bigm| \mathrm{O}\mathrm{u}\mathrm{t}(K/H)
\bigm| \bigm| . Moreover, \{ 313\} is a prime component of
K/H. By using Table 2, one can easily obtain that K/H \sim = L2(5
4) or O -
8 (5).
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CHARACTERIZATION BY ORDER AND DEGREE PATTERN OF THE SIMPLE GROUPS O -
8 (q) . . . 747
If K/H \sim = L2(5
4), then (| G/K| , 31) = 1 by
\bigm| \bigm| \mathrm{O}\mathrm{u}\mathrm{t}(L2(5
4))
\bigm| \bigm| = 8. Hence, the Sylow 31-subgroup
of H is of order 31 and is normal in G. Since (313, | \mathrm{A}\mathrm{u}\mathrm{t}(H31)| ) = 1, we deduce that G has an
element of order 31.313 by Lemma 2.9, which is a contradiction.
Therefore, we have K/H \sim = O -
8 (5). Because | G| = | O -
8 (5)| , we can get that | H| = 1, and,
thus, G \sim = O -
8 (5).
Proposition 3.4. The orthogonal group O -
8 (8) is OD-characterizable.
Proof. Suppose that G be a finite group such that | G| = | O -
8 (8)| = 236.37.5.73.13.17.19.73.241
and D(G) = (2, 6, 3, 6, 3, 1, 2, 2, 1). By Lemma 2.3, t(G) \geq 3. Since \mathrm{d}\mathrm{e}\mathrm{g}(2) = 2 and | \pi (G)| = 9,
it follows that t(2, G) \geq 2. Consequently, from Lemma 2.4 we implies that there exists a finite non-
Abelian simple group S such that S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), where K is the maximal normal solvable
subgroup of G.
We show that K is a p\prime -group, where p \in \{ 73, 241\} . Assume to the contrary that p \in \pi (K). Let
r \in \{ 13, 17, 19\} and r
\bigm| \bigm| | K| , then a Hall \{ p, r\} -subgroup of K is a cyclic group of order p.r, and,
hence, p is adjacent to r for all r \in \{ 13, 17, 19\} , which is a contradiction. Now, we may assume
that r /\in \pi (K). Let Kp \in \mathrm{S}\mathrm{y}\mathrm{l}p(K), then NG(Kp) contains an element of order r, so G contains
an element of order pr for all r \in \{ 13, 17, 19\} , which is again a contradiction. Therefore, K is a
\{ 73, 241\} \prime -group.
By Lemma 2.8, \pi (\mathrm{O}\mathrm{u}\mathrm{t}(S))\cap \{ 13, 41\} = \varnothing . On the other hand, since K is a \{ 73, 241\} \prime -group and
S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), it follows that the order of S is divisible by 73.241. According to the results
in Table 2, we obtain the only possibility for S is O -
8 (8). Therefore, O -
8 (8) \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(O -
8 (8)),
and since | G| = | O -
8 (8)| , we conclude that | K| = 1 and G \sim = O -
8 (8).
Proposition 3.5. The orthogonal group O -
8 (9) is OD-characterizable.
Proof. Let G be a finite group such that | G| = | O -
8 (9)| = 213.324.53.7.13.17.41.73.193 and
D(G) = (6, 3, 6, 3, 3, 1, 3, 2, 1). By Lemma 2.3, we have t(G) \geq 3. Furthermore, t(2, G) \geq 2
because of | \pi (G)| = 9 and \mathrm{d}\mathrm{e}\mathrm{g}(2) = 6. Therefore, Lemma 2.4 implies that there is a finite non-
Abelian simple group S such that S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), where K is the maximal normal solvable
subgroup of G.
We show that K is a p\prime -group, where p \in \{ 73, 193\} . By way of contradiction, let p \in \pi (K). If
r \in \{ 13, 17, 41\} , then, by using the same technique as in the proof of Propositions 3.4, we derive
that G has an element of order pr for all r \in \{ 13, 17, 19\} , which is impossible because \mathrm{d}\mathrm{e}\mathrm{g}(73) = 2
and \mathrm{d}\mathrm{e}\mathrm{g}(193) = 1. Therefore, K is a \{ 73, 193\} \prime -group.
From Lemma 2.8, we know that \pi (\mathrm{O}\mathrm{u}\mathrm{t}(S)) \cap \{ 73, 193\} = \varnothing . Since K is a \{ 73, 193\} \prime -group
and S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), it follows that the order of S is divisible by 73.193. Now, Table 2 shows
us that S is isomorphic to O -
8 (9). Since O -
8 (9) \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(O -
8 (9)) and | G| = | O -
8 (9)| , we
conclude that | K| = 1 and G \sim = O -
8 (9).
Proposition 3.6. The orthogonal group O -
8 (43) is OD-characterizable.
Proof. Let G be a finite group with | G| = | O -
8 (43)| = 210 \cdot 34 \cdot 52 \cdot 73 \cdot 113 \cdot 13 \cdot 17 \cdot
\cdot 37 \cdot 4312 \cdot 139 \cdot 193 \cdot 521 \cdot 631 and D(G) = (9, 9, 6, 9, 9, 9, 5, 2, 6, 5, 2, 2, 4). Since d1 = 2 and
d4 \leq | \pi (G)| - 5, then Lemma 2.2 implies that t(G) \geq 3. Moreover, t(2, G) \geq 2 because | \pi (G)| = 12
and \mathrm{d}\mathrm{e}\mathrm{g}(2) = 9. Thus, by Lemma 2.4, there exists a finite non-Abelian simple group S such that
S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S), where K is the maximal normal solvable subgroup of G.
We show that K is a p\prime -group, where p \in \{ 521, 631\} . Assume to the contrary that | K| is divisible
by p. If r \in \{ 13, 17, 37, 139, 193\} , then, by using a similar arguments as in the proof of Proposi-
tion 3.4, we can show that G has an element of order pr for all r \in \{ 13, 17, 37, 139, 193\} , which is
contradiction because \mathrm{d}\mathrm{e}\mathrm{g}(631) = 4 and \mathrm{d}\mathrm{e}\mathrm{g}(521) = 2. Therefore, K is a \{ 521, 631\} \prime -group.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
748 M. BIBAK, G. H. REZAEEZADEH
By Lemma 2.8, \mathrm{O}\mathrm{u}\mathrm{t}(S) is a \{ 521, 631\} \prime -group. Since S \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(S) and K is a
\{ 521, 631\} \prime -group, it follows that S is a simple group with \{ 521.631\} \subseteq \pi (S). Therefore, by using
Table 2 implies that S is isomorphic to O -
8 (43) and so O -
8 (43) \leq G/K \leq \mathrm{A}\mathrm{u}\mathrm{t}(O -
8 (43)). As
| G| = | O -
8 (43)| , we deduce that | K| = 1 and so G \sim = O -
8 (43).
Table 2. The orders of finite simple groups S \in Sp except alternating groups
S | S|
p = 41
L2(3
4) 24 \cdot 34 \cdot 5 \cdot 41
S4(9) 28 \cdot 38 \cdot 52 \cdot 17
Sz(32) 210 \cdot 52 \cdot 31 \cdot 41
L2(41) 23 \cdot 3 \cdot 5 \cdot 7 \cdot 41
O -
8 (3) 210 \cdot 312 \cdot 5 \cdot 7 \cdot 13 \cdot 41
L4(9) 210 \cdot 12 \cdot 52 \cdot 7 \cdot 13 \cdot 41
O9(3) 214 \cdot 316 \cdot 52 \cdot 7 \cdot 13 \cdot 41
S8(3) 214 \cdot 316 \cdot 52 \cdot 7 \cdot 13 \cdot 41
L2(41
2) 24 \cdot 3 \cdot 5 \cdot 7 \cdot 292 \cdot 412
S4(41) 28 \cdot 32 \cdot 52 \cdot 72 \cdot 292 \cdot 414
L2(2
10) 210 \cdot 3 \cdot 52 \cdot 11 \cdot 31 \cdot 41
S4(32) 220 \cdot 32 \cdot 52 \cdot 112 \cdot 312 \cdot 41
U5(4) 220 \cdot 32 \cdot 54 \cdot 13 \cdot 17 \cdot 41
O+
10(3) 215 \cdot 320 \cdot 52 \cdot 7 \cdot 112 \cdot 13 \cdot 41
U6(4) 230 \cdot 34 \cdot 56 \cdot 7 \cdot 132 \cdot 17 \cdot 41
p = 193
L2(3
8) 25 \cdot 38 \cdot 5 \cdot 17 \cdot 41 \cdot 193
S4(3
4) 210 \cdot 316 \cdot 52 \cdot 17 \cdot 412 \cdot 193
L2(193) 23 \cdot 3 \cdot 53 \cdot 97 \cdot 149 \cdot 1932
S4(193) 214 \cdot 32 \cdot 53 \cdot 972 \cdot 149 \cdot 149 \cdot 1934
U3(109) 24 \cdot 33 \cdot 52 \cdot 112 \cdot 61 \cdot 1093 \cdot 193
O -
8 (9) 213 \cdot 324 \cdot 53 \cdot 7 \cdot 13 \cdot 17 \cdot 41 \cdot 73 \cdot 193
L4(3
4) 213 \cdot 324 \cdot 53 \cdot 7 \cdot 13 \cdot 17 \cdot 412 \cdot 73 \cdot 193
S8(9) 218 \cdot 332 \cdot 54 \cdot 7 \cdot 13 \cdot 17 \cdot 412 \cdot 73 \cdot 193
O9(9) 218 \cdot 332 \cdot 54 \cdot 7 \cdot 13 \cdot 17 \cdot 412 \cdot 73 \cdot 193
O+
10(9) 220 \cdot 360 \cdot 54 \cdot 7 \cdot 112 \cdot 13 \cdot 17 \cdot 412 \cdot 67 \cdot 73 \cdot 193
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
CHARACTERIZATION BY ORDER AND DEGREE PATTERN OF THE SIMPLE GROUPS O -
8 (q) . . . 749
Table 2 (continued)
S | S|
p = 241
U3(16) 24.3.5.112.241
S8(8) 248.39.52.74.133.17.19.241
L2(2
12) 248.39.52.74.133.17.19.241
O -
8 (8) 236.37.5.73.13.17.19.73.241
L4(64) 236.37.52.73.132.17.19.73.241
3D4(4) 236.37.52.73.132.17.19.73.241
G2(16) 236.34.53.72.133.17.37.109.241
U4(64) 236.34.53.72.133.17.37.109.241
S6(64) 254.36.53.73.133.17.19.37.109.241
F4(8) 272.310.52.74.132.17.37.732.109.241
L3(2
12) 236.35.52.72.132.17.19.37.73.109.241
O+
8 (64) 272.37.53.74.134.172.37.73.109.2412
S4(64) 260.39.52.75.132.172.19.31.73.151.241
O+
10(8) 260.39.52.75.132.172.19.31.73.151.241
p = 257
L2(257) 28.3.43.257
L2(2
8) 28.3.5.17.257
S4(16) 216.32.52.172.257
U4(16) 224.32.52.173.241.257
O -
8 (4) 224.34.53.7.13.17.257
S8(4) 232.35.54.7.13.172.257
L2(241
2) 25.3.5.73.112.113.2412.257
S4(241) 210.32.52.114.113.2412.257
U3(257) 211.32.7.13.43.241.2573
O -
10(4) 240.35.56.7.13.172.41.257
L3(2
8) 224.32.52.7.13.172.241.257
S6(16) 236.34.53.7.13.173.241.257
O+
8 (16) 248.35.54.7.13.174.241.257
F4(4) 248.36.54.72.132.172.241.257
O+
10(4) 240.36.54.7.11.13.172.31.257
L5(16) 240.35.54.7.11.13.172.31.41.257
S10(4) 250.36.56.7.11.13.172.31.41.257
S20(2) 2100.314.56.73.112.13.172.19.312.41.43.73.127.257
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
750 M. BIBAK, G. H. REZAEEZADEH
Table 2 (continued)
S | S|
U10(4) 290.36.510.7.11.133.172.29.31.37.412.109.113.257
L7(16) 284.38.57.72.11.132.173.29.31.41.43.113.127.241.257
S14(4) 298.37.56.72.11.132.173.29.31.41.43.113.127.241.257
O+
16(4) 2112.38.57.72.11.132.174.29.31.41.43.113.127.241.2572
O+
22(2) 2110.314.56.73.112.13.172.19.23.312.41.43.73.89.127.257
E7(4) 2126.311.58.73.11.133.172.19.29.31.37.41.43.73.109.113.127.241.257
p = 313
L2(5
4) 26 \cdot 3 \cdot 54.13.313
S4(25) 29 \cdot 32 \cdot 58 \cdot 132 \cdot 313
O -
8 (5) 210.34.512.7.13.31.313
O9(5) 215 \cdot 35 \cdot 54.7 \cdot 132.31.313
S8(5) 215 \cdot 35 \cdot 54.7 \cdot 132.31.313
L4(25) 29 \cdot 34 \cdot 512.7.132.31.313
L3(313) 27 \cdot 34 \cdot 132 \cdot 157.1812.3133
L2(313
2) 26 \cdot 3 \cdot 5 \cdot 13.97.101.157.3132
S4(313) 29 \cdot 32.5 \cdot 13 \cdot 97.101.1572.3134
L4(313) 213 \cdot 34 \cdot 5 \cdot 133.97.101.1572.1812.3136
3D4(29) 26 \cdot 34 \cdot 52 \cdot 72.132.2912.37.61.672.2712.313
p = 631
L3(43) 24 \cdot 32 \cdot 72 \cdot 11 \cdot 433 \cdot 631
L2(43) 22 \cdot 32 \cdot 7 \cdot 11 \cdot 13 \cdot 433 \cdot 139 \cdot 631
L3(587) 24 \cdot 3 \cdot 72 \cdot 2932 \cdot 547 \cdot 5873 \cdot 631
L3(631) 25 \cdot 34 \cdot 52 \cdot 72 \cdot 79 \cdot 307 \cdot 433 \cdot 631
L4(43) 27 \cdot 34 \cdot 52 \cdot 73 \cdot 112 \cdot 37 \cdot 436 \cdot 631
G2(43) 26 \cdot 34 \cdot 72 \cdot 112 \cdot 13 \cdot 436 \cdot 139 \cdot 631
O+
8 (43 28 \cdot 34 \cdot 52 \cdot 73 \cdot 113 \cdot 37 \cdot 4312 \cdot 139 \cdot 631
S6(43) 29 \cdot 34 \cdot 52 \cdot 73 \cdot 113 \cdot 37 \cdot 439 \cdot 139 \cdot 631
O7(43) 29 \cdot 34 \cdot 52 \cdot 73 \cdot 113 \cdot 37 \cdot 439 \cdot 139 \cdot 631
L3(43
2) 27 \cdot 32 \cdot 52 \cdot 72 \cdot 112 \cdot 13 \cdot 37 \cdot 436 \cdot 139 \cdot 631
L4(43) 29 \cdot 34 \cdot 52 \cdot 73 \cdot 113 \cdot 13 \cdot 37 \cdot 436 \cdot 139 \cdot 631
O -
8 (43) 210 \cdot 34 \cdot 52 \cdot 73 \cdot 113 \cdot 13 \cdot 17 \cdot 37 \cdot 4312 \cdot 139 \cdot 193 \cdot 521 \cdot 631
S8(43) 214 \cdot 35 \cdot 54 \cdot 74 \cdot 114 \cdot 13 \cdot 372 \cdot 139 \cdot 193 \cdot 521 \cdot 631
O9(43) 214 \cdot 35 \cdot 54 \cdot 74 \cdot 114 \cdot 13 \cdot 372 \cdot 139 \cdot 193 \cdot 521 \cdot 631
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
CHARACTERIZATION BY ORDER AND DEGREE PATTERN OF THE SIMPLE GROUPS O -
8 (q) . . . 751
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ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
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| id | umjimathkievua-article-2357 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:25Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/97/dcf5d177a5ed1c8904b98d7b7a27e197.pdf |
| spelling | umjimathkievua-article-23572022-07-15T07:54:29Z Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ Bibak, M. Rezaeezadeh, G. H. Bibak, M. Rezaeezadeh, G. H. Prime Graph degree pattern OD-Characterizability Simple Group UDC 512.5 In this paper, it is demonstrated that every finite group $G$ with the same order and degree pattern as $O^{-}_{8}(q)$ for certain $q$ is necessarily isomorphic to the group $O^{-}_{8}(q)$. УДК 512.5 Характеристика порядкiв та степенiв простих груп $O^{-}_{8}(q)$ для заданого $q$ Доведено, що будь-яка скiнченна група $G$, яка має тi ж самi порядок та степiнь, що й група $O^{-}_{8}(q)$ для деякого $q$, необхiдно має збiгатися з $O^{-}_{8}(q)$. Institute of Mathematics, NAS of Ukraine 2022-07-07 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2357 10.37863/umzh.v74i6.2357 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 6 (2022); 743 - 751 Український математичний журнал; Том 74 № 6 (2022); 743 - 751 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2357/9247 Copyright (c) 2022 gholamreza rezaeezadeh |
| spellingShingle | Bibak, M. Rezaeezadeh, G. H. Bibak, M. Rezaeezadeh, G. H. Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title | Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title_alt | Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title_full | Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title_fullStr | Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title_full_unstemmed | Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title_short | Characterization by order and degree pattern of the simple groups $O^{-}_{8}(q)$ for certain $q$ |
| title_sort | characterization by order and degree pattern of the simple groups $o^{-}_{8}(q)$ for certain $q$ |
| topic_facet | Prime Graph degree pattern OD-Characterizability Simple Group |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2357 |
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