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1.
Gérard Endimioni 《Monatshefte für Mathematik》2009,156(1):23-29
Let G be a group and let n be a positive integer. A polynomial function in G is a function from G
n
to G of the form , where f(x
1, . . . , x
n
) is an element of the free product of G and the free group of rank n freely generated by x
1, . . . , x
n
. There is a natural definition for the product of two polynomial functions; equipped with this operation, the set of polynomial functions is a group. We prove that this group is polycyclic if and only if G is finitely generated, soluble, and nilpotent-by-finite. In particular, if the group of polynomial functions is polycyclic,
then necessarily it is nilpotent-by-finite. Furthermore, we prove that G itself is polycyclic if and only if the subgroup of polynomial functions which send (1, . . . , 1) to 1 is finitely generated
and soluble.
相似文献
2.
设A是秩为n(n≥2)的自由Abel群,A的自同构群Aut(A)=GL(n,Z).对整数m,取α=(010…000┆┆┆┆┆┆┆000…0110…0 m)记∈Aut(A).记Гm(n)=A×<α>则它是一个2元生成的多重循环群.本文给出了Γm(n)的准确的剩余有限性质. 相似文献
3.
A. Alves 《Topology》2006,45(1):1-25
We give an explicit formula for the Whitehead group of a three-dimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ. 相似文献
4.
We present an upper bound for |G| of a group G of even order possessing a unique conjugacy class of involutions. 相似文献
5.
A. G. Savushkina 《Mathematical Notes》1996,60(1):68-80
The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating
automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices
is sharpened.
Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996. 相似文献
6.
7.
On Connection Between the Structure of a Finite Group and the Properties of Its Prime Graph 总被引:1,自引:0,他引:1
A. V. Vasil’ev 《Siberian Mathematical Journal》2005,46(3):396-404
It is shown that the condition of nonadjacency of 2 and at least one odd prime in the Gruenberg-Kegel graph of a finite group G under some natural additional conditions suffices to describe the structure of G; in particular, to prove that G has a unique nonabelian composition factor. Applications of this result to the problem of recognition of finite groups by spectrum are also considered.Original Russian Text Copyright © 2005 Vasilev A. V.The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8294), the Program Universities of Russia (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 511–522, May–June, 2005. 相似文献
8.
It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to L3(3) or U3(3) (to U3(3) or S4(3), respectively). 相似文献
9.
The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973) and Moghaddam et al. (2005). Moreover, we shall define the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article. 相似文献
10.