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1.
We prove that a finite group, having the same set of element orders as a finite simple group L and the prime graph with at least three connected components, possesses a (unique) nonabelian composition factor which is isomorphic to L, unless L is isomorphic to the alternating group of degree 6. 相似文献
2.
O. A. Alekseeva 《Algebra and Logic》2006,45(1):1-11
It is proved that if G is a finite group with an element order set as in the simple group 3D4(q), where q is even, then the commutant of G/F(G) is isomorphic to 3D4(q) and the factor group G/G′ is a cyclic {2, 3}-group.
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Translated from Algebra i Logika, Vol. 45, No. 1, pp. 3–19, January–February, 2006. 相似文献
3.
It is proved that if L is one of the simple groups 3D4(q) or F4(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is
isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group.
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Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005.
Supported by RFBR grant No. 04-01-00463. 相似文献
4.
证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画. 相似文献
5.
A. V. Vasilyev 《Algebra and Logic》2002,41(2):74-80
It is proved that a finite group that is isomorphic to a simple non-Abelian group G=G
2(3
n
) is, up to isomorphism, recognized by a set (G) of its element orders, that is, H G if (H)=(G) for some finite group H. 相似文献
6.
V. D. Mazurov 《Algebra and Logic》2002,41(2):93-110
It is proved that among simple groups $S_4(q)$ in the class of finite groups, only the groups $S_4(3^n)$, where $n$ is an odd number greater than unity, are recognizable by a set of their element orders. It is also shown that simple groups $U_3(9)$, ${^3D}_4(2)$, $G_2(4)$, $S_6(3)$, $F_4(2)$, and ${^2E}_6(2)$ are recognizable, but $L_3(3)$ is not. 相似文献
7.
For G a finite group,π_e(G) denotes the set of orders of elements in G.If Ω is a subsetof the set of natural numbers,h(Ω) stands for the number of isomorphism classes of finite groups withthe stone set Ω of element orders.We say that G is k-distinguishable if h(π_e(G))=k<∞,otherwiseG is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.Itis shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M~cL and O'N,then Aut(M) is characterizable by its element orders.It is also proved that if M is isomorphic toM_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}. 相似文献
8.
For G a finite group, π e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M 12, M 22, J 2, He, Suz, M c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M 12, M 22, He, Suz or O′N, then h(π e (Aut(M))) ∈¸ {1,∞}. 相似文献
9.
It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to L3(3) or U3(3) (to U3(3) or S4(3), respectively). 相似文献
10.
A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L
n
(2k) are recognizable by spectrum for n = 2m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005. 相似文献
11.
For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime
graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set
with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders
on these sets; the information obtained is collected in tables. We consider several applications of these results to various
problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups.
Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science
Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher
School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant
No. 86-197.
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Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005. 相似文献
12.
13.
Seyyed Majid Jafarian Amiri 《代数通讯》2013,41(6):2055-2059
In [1], the authors proved that the maximum sum of element orders on finite groups of the same order occurs in cyclic group. In this paper we obtain the structure of groups having maximum sum of element orders on noncyclic nilpotent group of the same order. 相似文献
14.
Given a finite group G, write ψ(G) to denote the sum of the orders of the elements of G. Our main result is that if C is a cyclic group and G is a noncyclic group of the same order, then ψ(G) < ψ(C). 相似文献
15.
Ali Reza Ashrafi 《Algebra Colloquium》2000,7(2):139-146
For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A4, the alternating group on four letters. Also, we prove that, if G/Z(G) A4, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A4 and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07 相似文献
16.
Groups Saturated by a Finite Set of Groups 总被引:1,自引:1,他引:0
We study the periodic groups that are saturated by a finite set of finite (nonabelian simple) groups, obtaining some information about the elements of the saturating set. 相似文献
17.
Ming Chun XU 《数学学报(英文版)》2005,21(4):899-902
In this paper the following theorem is proved: Every group L3(q) for q = 3^(2m-1)(m≥2) is characterized by its set of element orders. 相似文献
18.
Let G be a finite group. Then we denote ψ(G) the sum of element orders in G. In [1] it is proved that if G is a non-cyclic group of order n, then ψ(G) < ψ(C n ), where C n is the cyclic group of order n. Here we generalize and improve this result, and also we give an application of this improvement. 相似文献
19.
Xianglin Du 《代数通讯》2013,41(4):1345-1359
ABSTRACT Let k(G) be the number of conjugacy classes of finite groups G and π e (G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |π e (G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A 5, L 2(7), S 5, Z 3, Z 4, S 4, A 4, D 10, Hol(Z 5), or Z 3 ? Z 4. 相似文献
20.
On Connection Between the Structure of a Finite Group and the Properties of Its Prime Graph 总被引:1,自引:0,他引:1
A. V. Vasil’ev 《Siberian Mathematical Journal》2005,46(3):396-404
It is shown that the condition of nonadjacency of 2 and at least one odd prime in the Gruenberg-Kegel graph of a finite group G under some natural additional conditions suffices to describe the structure of G; in particular, to prove that G has a unique nonabelian composition factor. Applications of this result to the problem of recognition of finite groups by spectrum are also considered.Original Russian Text Copyright © 2005 Vasilev A. V.The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8294), the Program Universities of Russia (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 511–522, May–June, 2005. 相似文献