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1.
V. D. Mazurov 《Algebra and Logic》1998,37(6):371-379
For G a finite group, ω(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 nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G
is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups
PGLn(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S4(7) is recognizable, whereas A10, U3(5), U3(7), U4(2), and U5(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of
its order is at most 11. Then one of the following holds: (i) G is isomorphic to A5, A7, A8, A9, A11, A12, L2(q), q=7, 8, 11, 49, L3(4), S4(7), U4(3), U6(2), M11, M12, M22, HS, or McL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A6, A10, U3(3), U4(2), U5(2), U3(5), or J2, and G is nonrecognizable; (iii) G is isomorphic to S6(2) or O
8
+
(2), and h(ω(G))=2.
Supported by RFFR grant No. 96-01-01893.
Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998. 相似文献
2.
A. K. Shlyopkin 《Algebra and Logic》1998,37(5):345-350
A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic
to some group of X. We study periodic conjugate biprimitive finite groups saturated with groups in the set {U3(2n)}. It is proved that every such group is isomorphic to a simple group U3(Q) over a locally finite field Q of characteristic 2.
Supported by the RF State Committee of Higher Education.
Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 606–615, September–October, 1998. 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.
It is proved that finite simple groups L4(2m), m ⩾ 2, and U4(2m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups.
As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order
8.
Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry
of Education of China (Project for Retaining Foreign Expert).
Supported by NSF of Chongqing (CSTC: 2005BB8096).
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Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008. 相似文献
5.
Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup
of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups
in a set {U3(2m) | m = 1, 2, …} is isomorphic to U3(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite.
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Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008. 相似文献
6.
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
7.
Cao H. P. Chen G. Grechkoseeva M. A. Mazurov V. D. Shi W. J. Vasil'ev A. V. 《Siberian Mathematical Journal》2004,45(6):1031-1035
The spectrum of a finite group is the set of its element orders. A finite group G is said to be recognizable by spectrum, if every finite group with the same spectrum as G is isomorphic to G. The purpose of the paper is to prove that for every natural m the finite simple Chevalley group F
4(2
m
) is recognizable by spectrum. 相似文献
8.
V. D. Mazurov 《Algebra and Logic》1997,36(1):23-32
For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω)
stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic
to S9, S11, S12, S13, or A12, and h(ω(G))=2 if G is isomorphic to S2(6) or to O
8
+
(2). 01
Supported by RFFR grant No. 96-01-01893.
Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 37–53, January–February, 1997. 相似文献
9.
A. I. Sozutov 《Algebra and Logic》2000,39(5):345-353
An involution i of a group G is said to be finite if |iig|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover,
the normalizer H=NG(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic
to L2(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the
Kourovka Notebook is answered in the affirmative.
Supported by RFFR grant No. 99-01-00542.
Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000. 相似文献
10.
A. S. Kondratev 《Siberian Mathematical Journal》2007,48(6):1001-1018
We prove that if L is one of the simple groups E
6(q) and 2
E
6(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group.
Original Russian Text Copyright ? 2007 Kondrat’ev A. S.
The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007. 相似文献
11.
As shown in [1] the simple group 2 D2m + 1 (3)^2 D_{2^m + 1} (3) is recognizable by spectrum. The main result of this paper generalizes the above, stating that 2 D2m + 1 (3)^2 D_{2^m + 1} (3) is recognizable by prime graph. In other words, we show that if G is a finite group satisfying G(G) = G(2 D2m + 1 (3))\Gamma (G) = \Gamma (^2 D_{2^m + 1} (3)) then G @ 2 D2m + 1 (3)G \cong ^2 D_{2^m + 1} (3). 相似文献
12.
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
13.
V. P. Burichenko 《Algebra and Logic》2008,47(6):384-394
Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S2(V) be its symmetric square. We construct a 2-cohomology group H2(G, L). The group is one-dimensional over F
q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H2(G, L) = 0. Previously, such groups H2(G, L) were known for the cases where n = 2 or q = p is prime. We state that H2(G, L) are trivial for n ⩾ 3 and q = pm, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods.
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Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008. 相似文献
14.
LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G,
S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S)
is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ4×ℤ2 orG≅Q
8×ℤ
2
r
(r⩾0) and that every finite group has a normal Cayley digraph, where Zm is the cyclic group of orderm and Q8 is the quaternion group of order 8.
Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China. 相似文献
15.
E. P. Vdovin 《Algebra and Logic》2000,39(5):301-312
Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved
that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|2<|G|.
Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program
for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000.
Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000. 相似文献
16.
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) = r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B
p
(3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B
p
(3)), where p > 3 is an odd prime, then G @ Bp(3){G\cong B_p(3)} or C
p
(3). Also if Γ(G) = Γ(B
3(3)), then G @ B3(3), C3(3), D4(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O2(G) @ Aut(2B2(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained. 相似文献
17.
V. E. Kislyakov 《Algebra and Logic》1996,35(5):305-309
We study the problem as to which is the cardinality of connected components of the graph Γα, defined as follows. Let G be a group and a an element of G. The vertex set V(Γα) of the graph is the conjugacy class of elements,Cl
G(a), and two vertices x and y of the graph Γα are bridged by an edge iff x=y. If the intersectionC
G(a)∩Cl
G(a) is finite, Γα is locally finite. We prove that connected components of the locally finite graph Γα are finite in some classes of groups.
Supported by RFFR grant No. 94-01-01084.
Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996. 相似文献
18.
Let
be a set of finite groups. A group G is saturated with groups from
if every finite subgroup of G is contained in a subgroup isomorphic to some member of
. It is proved that a periodic group G saturated with groups from the set {L3(2m)|m = 1, 2, …} is isomorphic to L3(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite.
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Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007. 相似文献
19.
V. A. Roman'kov 《Algebra and Logic》2007,46(1):37-45
It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged
Dehn function σ(n) is subasymptotic w.r.t. the function nl+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite
rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic.
Supported by RFBR grant No. 04-01-00489.
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Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007. 相似文献
20.
We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the
p-rank of Ext
ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable
strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π0 and Π1, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2μ = μ+ such that the p-rank of Ext
ℤ(G, ℤ) equals 2μ = μ+ for every p ∈ Π0 and 0 otherwise, that is, for p ∈ Π1.
Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development
project No. I-706-54.6/2001.
Supported by a grant from the German Research Foundation DFG.
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Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007. 相似文献