共查询到20条相似文献,搜索用时 15 毫秒
1.
It is an open question whether or not every finitep-groupG has a presentation withd(G)=dimH
1(G,Z
p
) generators andr(G)=dimH
2(G, Z
p
) relations; in this article, a large number of examples are given to show that such a presentation does exist for nearly
all such groups for whichr(G) has been calculated. 相似文献
2.
IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let
be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH
1(G,z
p
) generators andr(G)=dimH
2 (G,Z
p
) relations: in this article it is shown that ifK is a non cyclic group belonging to
andH is a finite abelian p-group then any outer extension ofK byH belongs to
. 相似文献
3.
Dimension subgroups in characteristicp are employed in the study of the power structure of finitep-groups. We show, e.g., that ifG is ap-group of classc (p odd) andk=⌜log
p
((c+1)/(p−1))⌝, then, for alli, any product ofp
i+k
th powers inG is ap
i
th power. This sharpens a previous result of A. Mann. Examples are constructed in order to show that our constantk is quite often the best possible, and in any case cannot be reduced by more than 1.
Partially supported by MPI funds. This author is a member of GNSAGA-CNR.
Partially supported by a Rothschild Fellowship. 相似文献
4.
Letk be any field andG a finite group. Given a cohomology class α∈H
2(G,k
*), whereG acts trivially onk
*, one constructs the twisted group algebrak
αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z
n×Zn). This paper has two main results: First we prove that ifD=k
α
G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and
, the division algebraD=k
α
G is a product of cyclic algebras. Furthermore, ifD
p is ap-primary factor ofD, thenD
p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and
, the same result holds forD
p, p odd. Ifp=2 we show thatD
2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z
2×Z2n. 相似文献
5.
LetA be an elementary abelianq-group acting on a finiteq′-groupG. We show that ifA has rank at least 3, then properties ofC
G(a)′, 1 ≠a ∈A restrict the structure ofG′. In particular, we consider exponent, order, rank and number of generators.
This author was supported by the NSF.
This author was supported by CNPq-Brazil. 相似文献
6.
Peter Danchev 《Rendiconti del Circolo Matematico di Palermo》2002,51(3):391-402
LetF be a field of characteristicp>0 and letG be an arbitrary abelian group written multiplicatively withp-basis subgroup denoted byB. The first main result of the present paper is thatB is an isomorphism invariant of theF-group algebraFG. In particular, thep-local algebraically compact groupG can be retrieved fromFG. Moreover, for the lower basis subgroupB 1 of thep-componentG p it is shown thatG p/Bl is determined byFG. Besides, ifH is (p-)high inG, thenG p/Hp andH p n[p] for ℕ0 are structure invariants forFG, andH[p] as a valued vector space is a structural invariant forN 0 G, whereN p is the simple field ofp-elements. Next, presume thatG isp-mixed with maximal divisible subgroupD. ThenD andF(G/D) are functional invariants forFG. The final major result is that the relative Ulm-Kaplansky-Mackeyp-invariants ofG with respect to the subgroupC are isomorphic invariants of the pair (FG, FC) ofF-algebras. These facts generalize and extend analogous in this aspect results due to May (1969), Berman-Mollov (1969) and Beers-Richman-Walker (1983). As a finish, some other invariants for commutative group algebras are obtained. 相似文献
7.
Giuseppe Molteni 《Archiv der Mathematik》2002,79(6):432-438
We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H
b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF
p. This implies that the equationF
a=Gb with (a, b)=1 has the unique solutionF=H
b andG=H
a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two. 相似文献
8.
W. O. Alltop 《Israel Journal of Mathematics》1976,23(1):31-38
ItH
i
is a finite non-abelianp-group with center of orderp, for 1≦j≦R, then the direct product of theH
i
does not occur as a normal subgroup contained in the Frattini subgroup of any finitep-group. If the Frattini subgroup Φ of a finitep-groupG is cyclic or elementary abelian of orderp
2, then the centralizer of Φ inG properly contains Φ. Non-embeddability properties of products of groups of order 16 are established. 相似文献
9.
Thomas Meixner 《Israel Journal of Mathematics》1985,51(1-2):68-78
For a (finite) groupG and some prime powerp
n, theH
p
n
-subgroupH
pn (G) is defined byH
p
n
(G)=〈xεG|x
pn≠1〉. A groupH≠1 is called aH
p
n
-group, if there is a finite groupG such thatH is isomorphic toH
p
n
(G) andH
p
n
(G)≠G. It is known that the Fitting length of a solvableH
p
n
-group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH
p
n
-groups having Fitting lengthn, this result is “almost” best possible. 相似文献
10.
A graphH divides a graphG, writtenH|G, ifG isH-decomposable. A graphG without isolated vertices is a greatest common divisor of two graphsG
1 andG
2 ifG is a graph of maximum size for whichG|G
1 andG|G
2, while a graphH without isolated vertices is a least common multiple ofG
1 andG
2 ifH is a graph of minimum size for whichG
1|H andG
2|H. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple ofG
1 andG
2 to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairsG
1,G
2 of graphs are determined, including when one ofG
1 andG
2 is a cycle of even length and the other is a star.G. C's research was supported in part by the Office of Naval Research, under Grant N00014-91-I-1060 相似文献
11.
H. Markšaitis 《Lithuanian Mathematical Journal》2000,40(1):39-47
LetK
p (p, q) be the maximalp-extension of the field ℚ of rational numbers with ramification pointsp andq. LetG
p (p, q) be the Galois group of the extensionK
p(p.q)/ℚ. It is known thatG
p(p, q) can be presented by two generators which satisfy a single relation. The form of this relation is known only modulo
the second member of the descending central series ofG
p(p, q). In this paper, we find an arithmetical-type condition on which the form of the relation modulo the third member of
the descending central series ofG
p(p, q) depends. We also consider two examples withp=3,q=19 andp=3,q=37.
Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 48–60, January–March, 2000.
Translated by H. Markšaitis 相似文献
12.
SupposeG is a finitep-group andk is the field ofp elements, and letU be the augmentation ideal of the group algebrakG. We investigate whichp-groups,G, have normal complements in their modp-envelope,G
*.G
* is defined byG
*={1−u∶u∈U}. 相似文献
13.
B. A. F. Wehrfritz 《Israel Journal of Mathematics》1984,47(2-3):154-164
LetD=F(G) be a division ring generated as a division ring by its central subfieldF and the polycyclic-by-finite subgroupG of its multiplicative group, letn be a positive integer and letX be a finitely generated subgroup of GL(n, D). It is implicit in recent works of A. I. Lichtman thatX is residually finite. In fact, much more is true. If charD=p≠0, then there is a normal subgroup ofX of finite index that is residually a finitep-group. If charD=0, then there exists a cofinite set π=π(X) of rational primes such that for eachp in π there is a normal subgroup ofX of finite index that is residually a finitep-group. 相似文献
14.
Gabriel Navarro 《Israel Journal of Mathematics》2002,132(1):277-283
Suppose thatG is a finitep-solvable group and letPε Syl
p
(G). In this note, we prove that the character table ofG determines ifN
G(itP)/P is abelian.
Research partially supported by DGICYT. 相似文献
15.
LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD
w
.D
1 is the only invariant control set, and the subsetW(S)={w:D
w
=D
1} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold.
Research partially supported by CNPq grant no 50.13.73/91-8 相似文献
16.
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
17.
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
18.
Pavel Shumyatsky 《Israel Journal of Mathematics》1994,87(1-3):111-116
Letp be a prime,G a periodic solvablep′-group acted on by an elementary groupV of orderp
2. We show that ifC
G(v) is abelian for eachv ∈V
# thenG has nilpotent derived group, and ifp=2 andC
G(v) is nilpotent for eachv ∈V
# thenG is metanilpotent. Earlier results of this kind were known only for finite groups. 相似文献
19.
LetG be a finitep-group, and letU(G) be the group of units of the group algebraFG, whereF is a field of characteristicp. It is shown that, if the commutative subgroup ofG has order at leastp
2, then the nilpotency class ofU(G) is at least 2p−1.
The authors are grateful to the Dipartimento di Matematica of the Universita di Trento, and to the Mathematical Institute
of the University of Oxford, for their hospitality while this paper was being written. Then are also grateful to Robert Sandling,
for communication of results, and problems, prior to publication. 相似文献
20.
Angel Carocca 《Bulletin of the Brazilian Mathematical Society》1995,26(2):161-165
Given a finite groupG andp an odd prime number, we conclude thatO
p(G)G isp-nilpotent when for every subgroupH ofG of orderp there exists a subgroupK ofG such thatG=HK andH permutes with every subgroup ofK. 相似文献