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1.
Enrico Jabara 《Rendiconti del Circolo Matematico di Palermo》2007,56(3):343-348
LetG be a group and ϕ an automorphism ofG. Two elementsx, y ∈ G are called ϕ-conjugate if there existsg ∈ G such thatx=g
−1
yg
θ. It is easily verified that the ϕ-conjugation is an equivalence relation; the numberR(ϕ) of ϕ-classes ofG is called the Reidemeister number of the automorphism ϕ.
In this paper we prove that if a polycyclic groupsG admits an automorphism ϕ of ordern such thatR(ϕ)<∞, thenG contains a subgroup of finite index with derived length at most 2
n−1
.
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2.
LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev.
This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars
(No. 10425103) and the Key Program of NSF (No. 10331020) in China. 相似文献
3.
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
4.
Edward A. Bertram 《Israel Journal of Mathematics》1984,47(4):335-344
In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C
G(x)|>|G|1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C
G(x)|>[G:G′∩Z]1/2 (and thus|C
G(x)|>|G|1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C
G(x)|>|G|1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp
mqn (p, q primes) contains a proper centralizer of order >|G|1/2. Finally, in Theorem 3 we show that theaverage
|C(x)|, x∈G, is ≧c|G|
1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible. 相似文献
5.
LetG be a finite group, andS a subset ofG \ |1| withS =S
−1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS
α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S
−1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA
5 is a 4-Cl-group, which was a conjecture posed by Li and Praeger. 相似文献
6.
Enrico Jabara 《Rendiconti del Circolo Matematico di Palermo》2005,54(3):367-380
An automorphismϕ of a groupG is said to be uniform il for everyg ∈G there exists anh ∈G such thatG=h
−1
h
ρ
. It is a well-known fact that ifG is finite, an automorphism ofG is uniform if and only if it is fixed-point-free. In [7] Zappa proved that if a polycyclic groupG admits an uniform automorphism of prime orderp thenG is a finite (nilpotent)p′-group.
In this paper we continue Zappa’s work considering uniform automorphism of orderpg (p andq distinct prime numbers). In particular we prove that there exists a constantμ (depending only onp andq) such that every torsion-free polycyclic groupG admitting an uniform automorphism of orderpq is nilpotent of class at mostμ. As a consequence we prove that if a polycyclic groupG admits an uniform automorphism of orderpq thenZ
μ
(G) has finite index inG.
Al professore Guido Zappa per il suo 900 compleanno 相似文献
7.
Martyn R. Dixon Martin J. Evans Antonio Tortora 《Central European Journal of Mathematics》2010,8(1):22-25
A subgroup H of a group G is inert if |H: H ∩ H
g
| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally
graded simple groups cannot be totally inert. 相似文献
8.
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
9.
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
10.
Let G be a finite group written multiplicatively and k a positive integer. If X is a non-empty subset of G, write X
2 = |xy | x, y X . We say that G has the small square property on k-sets if |X
2| < k
2 for any k-element subset X of G. For each group G, there is a unique m = m
G
such that G has the small square property on (m + 1)-sets but not on m-sets. In this paper we show that given any positive integer d, there is a finite group G with m
G
= d. 相似文献
11.
Enrico Jabara 《Rendiconti del Circolo Matematico di Palermo》2001,50(3):393-404
LetG be a group and α an automorphism ofG; α is calledn-splitting if
for allg∈G. In this note we study the structure of finite groups admitting an-splitting automorphism of order 2.
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12.
A Cayley graph Cay(G,S) of a groupGis called a CI-graph if wheneverTis another subset ofGfor which Cay(G,S) Cay(G,T), there exists an automorphism σ ofGsuch thatSσ = T. For a positive integerm, the groupGis said to have them-CI property if all Cayley graphs ofGof valencymare CI-graphs; further, ifGhas thek-CI property for allk ≤ m, thenGis called anm-CI-group, and a |G|-CI-groupGis called a CI-group. In this paper, we prove that Ais not a 5-CI-group, that SL(2,5) is not a 6-CI-group, and that all finite 6-CI-groups are soluble. Then we show that a nonabelian simple group has the 4-CI property if and only if it is A5, and that no nonabelian simple group has the 5-CI property. Also we give nine new examples of CI-groups of small order, which were found to be CI-groups with the assistance of a computer. 相似文献
13.
A linear automorphism of a finite dimensional real vector spaceV is calledproximal if it has a unique eigenvalue—counting multiplicities—of maximal modulus. Goldsheid and Margulis have shown that if a subgroupG of GL(V) contains a proximal element then so does every Zariski dense subsemigroupH ofG, providedV considered as aG-module is strongly irreducible. We here show thatH contains a finite subsetM such that for everyg∈GL(V) at least one of the elements γg, γ∈M, is proximal. We also give extensions and refinements of this result in the following directions: a quantitative version
of proximality, reducible representations, several eigenvalues of maximal modulus.
Partially supported by NSF grant DMS 9204-720. 相似文献
14.
LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K. 相似文献
15.
A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX
g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper.
Dedicated to the memory of our friend and collaborator Richard E. Phillips 相似文献
16.
Thomas Meixner 《Israel Journal of Mathematics》1981,38(4):345-360
For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms,
however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp
2 possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced.
Several examples ofp-groups having large power automorphism groups are given. 相似文献
17.
Ursula Hamenstädt 《Geometric And Functional Analysis》2009,19(1):170-205
Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H2cb(G, Lp(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).
Partially supported by Sonderforschungsbereich 611. 相似文献
18.
In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G X is the maximal subspace so that G⊥ = {x* ∈ X*|x*(y) = 0; y∈ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω,X). 相似文献
19.
L. G. Kovács 《Israel Journal of Mathematics》1988,63(1):119-127
LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT
m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT
m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally
isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic.
GivenT
m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal
subgroups are abstractly isomorphic toT
m. Secondly, form the direct productS
m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS
m×OutT whose projections inS
m are primitive. The second result of the paper is that there is a bijection between these two sets.
The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which
can fall into a single abstract isomorphism class. 相似文献
20.
Aner Shalev 《Israel Journal of Mathematics》1993,82(1-3):395-404
LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr.
To Professor John Thompson, in honor of his outstanding achievements 相似文献