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1.
A subgroup H of a finite group G is called a TI-subgroup if H ∩ H
x
= 1 or H for any x ∈ G. In this short note, the finite groups all of whose nonabelian subgroups are TI-subgroups are classified. 相似文献
2.
A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H
x
in 〈H, H
x
〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups. 相似文献
3.
Pavel Shumyatsky 《Monatshefte für Mathematik》2002,137(2):157-165
The following result is proved. Let G be a residually finite group satisfying the identity ([x
1, x
2][x
3, x
4])
n
≡ 1 for a positive integer n that is not divisible by p
2
q
2 for any distinct primes p and q. Then G′ is locally finite.
Received 7 May 2001; in revised form 3 December 2001 相似文献
4.
For a finite poset P = (V, ≤ ), let _s(P){\cal B}_s(P) consist of all triples (x,y,z) ∈ V
3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let Bs(G){\cal B}_s(G) consist of all triples (x,y,z) ∈ V
3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations Bs(P){\cal B}_s(P) and Bs(G){\cal B}_s(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation Bs(P)=Bs(G){\cal B}_s(P)={\cal B}_s(G). 相似文献
5.
Françcoise Point 《Archive for Mathematical Logic》2001,40(7):525-529
We show that if G is a group of finite Morley rank, then the verbal subgroup <w(G)> is of finite width, where w is a concise word. As a byproduct, we show that if G is any abelian-by-finite group, then G
n
=<x
n
(G)> is definable.
Received: 15 March 1996 / Published online: 18 July 2001 相似文献
6.
An involution v of a group G is said to be finite (in G) if vv
g
has finite order for any g ∈ G. A subgroup B of G is called a strongly embedded (in G) subgroup if B and G\B contain involutions, but B ∩ B
g
does not, for any g ∈ G\B. We prove the following results. Let a group G contain a finite involution and an involution whose centralizer in G is periodic. If every finite subgroup of G of even order is contained in a simple subgroup isomorphic, for some m, to L
2(2
m
) or Sz(2
m
), then G is isomorphic to L
2(Q) or Sz(Q) for some locally finite field Q of characteristic two. In particular, G is locally finite (Thm. 1). Let a group G contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in G is a 2-group, and every finite subgroup of even order in G is contained in a finite non-Abelian simple subgroup of G, then G is isomorphic to L
2(Q) or Sz(Q) for some locally finite field Q of characteristic two (Thm. 2).
Supported by RFBR (project No. 08-01-00322), by the Council for Grants (under RF President) and State Aid of Leading Scientific
Schools (grant NSh-334.2008.1), and by the Russian Ministry of Education through the Analytical Departmental Target Program
(ADTP) “Development of Scientific Potential of the Higher School of Learning” (project Nos. 2.1.1.419 and 2.1.1./3023). (D.
V. Lytkina and V. D. Mazurov)
Translated from Algebra i Logika, Vol. 48, No. 2, pp. 190–202, March–April, 2009. 相似文献
7.
Let F n be the free group of rank n, and let Aut+(F n ) be its special automorphism group. For an epimorphism π : F n → G of the free group F n onto a finite group G we call the standard congruence subgroup of Aut+(F n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ+(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ+(G, π) ≤ Aut+(F 2) has infinite abelianization. 相似文献
8.
N. S. Romanovskii 《Algebra and Logic》2009,48(2):147-160
A group G is said to be rigid if it contains a normal series of the form G = G
1 > G
2 > … > G
m
> G
m + 1 = 1, whose quotients G
i
/G
i + 1 are Abelian and are torsion free as right Z[G/G
i
]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group
uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x
1, …, x
n
over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being
Noetherian on G
n
, which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed.
It is proved that every rigid group is equationally Noetherian.
Supported by RFBR (project No. 09-01-00099) and by the Russian Ministry of Education through the Analytical Departmental Target
Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1.419).
Translated from Algebra i Logika, Vol. 48, No. 2, pp. 258–279, March–April, 2009. 相似文献
9.
LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups.
Research of the first author is supported by the NNSF of China (Grant No. 10171086) and QLGCF of Jiangsu Province and a Croucher
Fellowship of Hong Kong.
Research of the second author is partially supported by a UGC (HK) grant #2060176 (2001/2002). 相似文献
10.
Thomas P. Wakefield 《Algebras and Representation Theory》2011,14(4):609-623
Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups 2
G
2(q
2) for q
2 = 32m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many
of the nonabelian simple groups. 相似文献
11.
A subgroup H of a group G is called s-conditionally permutable in G if for every Sylow subgroup T of G there exists an element x ∈ G such that HTx = TxH. Using the concept of s-conditionally permutable subgroups, some new characterizations of finite groups are obtained and several interesting results are generalized. 相似文献
12.
Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |Q
G
|) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H ≦ H
τG
, where H
τG
is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let X ≦ E be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ. 相似文献
13.
Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {x
p
: p∈P}, and the set of relations is {x
p
⋅x
q
=x
p
: p≤q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤
B
|G) is well-founded. A well-generated algebra is superatomic.
THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated.
The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements.
THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B.
This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra.
Remember that an interval algebra is a special case of a poset algebra.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
14.
The purpose of this work is to obtain the commutator relations and Frobenius relations in a relatively free algebra F
(l) specified by the identity [x
1
, . . . , x
l
] = 0 over a field of characteristic p > 0. These relations for l > 3 are analogous to the relations in the algebra F
(3) and are applied to the T-spaces in the algebra F
(l). In order to study the relations in F
(l) in more detail, we construct a model algebra analogous to the Grassmann algebra. 相似文献
15.
Shi Rong Li 《数学学报(英文版)》2008,24(4):647-654
Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown: (1) G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; (2) G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F(H), the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G. 相似文献
16.
Marco Reni 《Monatshefte für Mathematik》1999,128(1):23-33
If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial
element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds.
Received 25 September 1997; in revised form 2 June 1998 相似文献
17.
A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C
G
(x)| = |C
G
(y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size
separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis
has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have
distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes.
Received: 10 April 2007 相似文献
18.
Let F be a partially ordered set (poset). A poset P is called F-free if P contains no subposet isomorphic to F. A finite poset F is said to have the maximal element property if every maximal F-free subposet of any finite poset P contains a maximal element of P. It is shown that a poset F with at least two elements has the maximal element property if and only if F is an antichain or F ≅ 2 + 2. 相似文献
19.
We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ⩽ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S
4-free and every minimal subgroup of P ∩ G
N
is c-supplemented in N
G
(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized. 相似文献
20.
Gejza Jenča 《Order》2010,27(1):41-61
We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element
below x by x
↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x
↓,x] is a subset of B. For every meager element (that means, an element x with x
↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect
algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered
effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2
M(E) given by h(a) = [0,a] ∩ M(E). 相似文献