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1.
We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gn with respect to each generating set is at most n(|G|?rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gn such that the diameter of Gn with respect to 𝒜 is at most n(|G|?rank(G)). We will establish evidence for each of the above mentioned conjectures. 相似文献
2.
V. A. Tolstykh 《Siberian Mathematical Journal》2006,47(5):950-954
A group is said to have finite width whenever it has finite width with respect to each inverse-closed generating set. Bergman showed [1] that infinite symmetric groups have finite width and asked whether the automorphism groups of several classical structures have finite width, mentioning in particular infinite dimensional general linear groups over fields. In this article we prove that infinite dimensional general linear groups over arbitrary division rings have finite width. We consider the problem of finite width for other infinite dimensional classical groups. 相似文献
3.
§1. BasicDefinitionsLetXandYbesets,foragivingmappingf:X→Y,x|→y=f(x),amappingfonpowersetcanbeconductedbyf:f:P(X)→P(Y),A|→B=f(A)={y|x∈A,y=f(x)}. Notethatabinaryoperationisaspecialmappingonsets.GivenabinaryoperationonsetXsuchthat::X×X→X, (x,y)|→x… 相似文献
4.
Alex Chigogidze Karl H. Hofmann John R. Martin 《Transactions of the American Mathematical Society》1997,349(11):4537-4554
Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.
5.
M. G. Amaglobeli 《Siberian Mathematical Journal》2007,48(1):3-7
We give a complete classification of the algebraic sets and coordinate groups for the systems of equations in one variable over a free nilpotent group. 相似文献
6.
Mark Pedigo 《代数通讯》2013,41(11):4462-4475
In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions. 相似文献
7.
Mark A.M. Lynch 《International Journal of Mathematical Education in Science & Technology》2013,44(4):540-544
In this article, a simple exercise for finding groups isomorphic to the symmetry group of the real line is presented. A mechanism for producing metrics on the real line is used to construct the group elements, and these transformations, by construction, are isometries with respect to the generating metric. 相似文献
8.
9.
《Discrete Mathematics》2020,343(3):111763
We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal 1. Also, we compute the average number of geodesics representing an element of given word-length in such groups. 相似文献
10.
N. S. Romanovskii 《Algebra and Logic》2007,46(4):274-280
The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases:
(1) G = Fn is a free metabelian group of rank n; (2) G = Wn,k is a wreath product of free Abelian groups of ranks n and k.
Supported by RFBR grant No. 05-01-00292.
__________
Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007. 相似文献
11.
The problem of computing the automorphism groups of an elementary Abelian Hadamard difference set or equivalently of a bent function seems to have attracted not much interest so far. We describe some series of such sets and compute their automorphism group. For some of these sets the construction is based on the nonvanishing of the degree 1-cohomology of certain Chevalley groups in characteristic two. We also classify bent functions f such that Aut(f) together with the translations from the underlying vector space induce a rank 3 group of automorphisms of the associated symmetric design. Finally, we discuss computational aspects associated with such questions. 相似文献
12.
Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (22d+2, 22d+1±2
d
, 22d
±2
d
). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2
d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 22d+2 with exponent 2
d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case. 相似文献
13.
The notion of self-similarity in the sense of iterated function system (IFS) for compact topological groups is given by ?. Koçak in Definition 3. In this work, first we give the definition of strong self-similar group in the sense of IFS. Then, we investigate the main properties of these groups. We also obtain the relations between profinite groups and strong self-similar groups in the sense of IFS. Finally, we construct some examples of these groups. 相似文献
14.
N. G. Khisamiev 《Algebra and Logic》2002,41(4):274-283
Let G be a completely decomposable torsion-free Abelian group and G= Gi, where G
i
is a rank 1 group. If there exists a strongly constructive numbering of G such that (G,) has a recursively enumerable sequence of elements g
i
G
i
, then G is called a strongly decomposable group. Let pi, i, be some sequence of primes whose denominators are degrees of a number p
i
and let
. A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that
for some numbers i
1,...,i
k
. We bring in the concept of a quasihyperhyperimmune set, and specify a necessary and sufficient condition on the characteristic of A subject to which the group in question is strongly decomposable. Also, it is proved that every hyperhyperimmune set is quasihyperhyperimmune, the converse being not true. 相似文献
15.
New (96,20,4)-symmetric design has been constructed, unique under the assumption of an automorphism group of order 576 action. The correspondence between a (96,20,4)-symmetric design having regular automorphism group and a difference set with the same parameters has been used to obtain difference sets in five nonabelian groups of order 96. None of them belongs to the class of groups that allow the application of so far known construction (McFarland, Dillon) for McFarland difference sets.AMS lassification: 05B05 相似文献
16.
J. E. Iiams 《Designs, Codes and Cryptography》2001,23(1):75-80
Nontrivial difference sets in 2-groups are part of the family of Hadamarddifference sets. An abelian group of order 22d+2 has a difference setif and only if the exponent of the group is less than or equal to2
d+2. We provide an exponent bound for a more general type of 2-groupwhich has a Hadamard difference set. A recent construction due to Davis and Iiamsshows that we can attain this bound in at least half of the cases. 相似文献
17.
18.
Tanja Vu
i
i 《组合设计杂志》2000,8(4):291-299
Six nonisomorphic new symmetric designs with parameters (100, 45, 20) are constructed by action of the Frobenius group E25 · Z12. This group proves to be their full automorphism group. Its Frobenius subgroup of order 100 acts on the designs as their nonabelian Singer group. The result is presented through six nonisomorphic new nonabelian (100, 45, 20) difference sets as well. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 291–299, 2000 相似文献
19.
Wenjun Xiao 《Discrete Applied Mathematics》2006,154(11):1640-1644
In this note we obtain a simple expression of any finite group by means of its generating set. Applying this result we partly solve a conjecture on diameters of Cayley graphs proposed by Babai and Seress. We also obtain some other conclusions on diameters on Cayley graphs. 相似文献
20.
G是一个群,I是一个指标集.令CG=G×I={(g,i):g∈G,i∈I};(a,i)(b,j)=(ab,k)with k=min{i,j}则CG是一个半群.事实上,CG是Clifford半群,并且CG代表了一类特殊的Clifford半群. 相似文献