scholarly journals SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

2016 ◽  
Vol 102 (1) ◽  
pp. 136-149 ◽  
Author(s):  
PETER M. NEUMANN ◽  
CHERYL E. PRAEGER ◽  
SIMON M. SMITH
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Finite Rank ◽  
Minimal Condition ◽  
Permutation Groups ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Second Variation ◽  
Formula Group ◽  
Finite Linear

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

scholarly journals Invariant Relations and Aschbacher Classes of Finite Linear Groups

10.37236/712 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jing Xu ◽  
Michael Giudici ◽  
Cai Heng Li ◽  
Cheryl E. Praeger
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Permutation Group ◽  
Cartesian Product ◽  
Permutation Groups ◽  
Linear Groups ◽  
Natural Action ◽  
Fourth Author ◽  
Empty Subset ◽  
Finite Linear

For a positive integer $k$, a $k$-relation on a set $\Omega$ is a non-empty subset $\Delta$ of the $k$-fold Cartesian product $\Omega^k$; $\Delta$ is called a $k$-relation for a permutation group $H$ on $\Omega$ if $H$ leaves $\Delta$ invariant setwise. The $k$-closure $H^{(k)}$ of $H$, in the sense of Wielandt, is the largest permutation group $K$ on $\Omega$ such that the set of $k$-relations for $K$ is equal to the set of $k$-relations for $H$. We study $k$-relations for finite semi-linear groups $H\leq{\rm\Gamma L}(d,q)$ in their natural action on the set $\Omega$ of non-zero vectors of the underlying vector space. In particular, for each Aschbacher class ${\mathcal C}$ of geometric subgroups of ${\rm\Gamma L}(d,q)$, we define a subset ${\rm Rel}({\mathcal C})$ of $k$-relations (with $k=1$ or $k=2$) and prove (i) that $H$ lies in ${\mathcal C}$ if and only if $H$ leaves invariant at least one relation in ${\rm Rel}({\mathcal C})$, and (ii) that, if $H$ is maximal among subgroups in ${\mathcal C}$, then an element $g\in{\rm\Gamma L}(d,q)$ lies in the $k$-closure of $H$ if and only if $g$ leaves invariant a single $H$-invariant $k$-relation in ${\rm Rel}({\mathcal C})$ (rather than checking that $g$ leaves invariant all $H$-invariant $k$-relations). Consequently both, or neither, of $H$ and $H^{(k)}\cap{\rm\Gamma L}(d,q)$ lie in ${\mathcal C}$. As an application, we improve a 1992 result of Saxl and the fourth author concerning closures of affine primitive permutation groups.

scholarly journals Quotient complete intersections of affine spaces by finite linear groups

1985 ◽  
Vol 98 ◽  
pp. 1-36 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Normal Subgroup ◽  
Affine Space ◽  
Finite Subgroup ◽  
Complete Intersections ◽  
Linear Groups ◽  
Coordinate Ring ◽  
Affine Spaces ◽  
Quotient Variety ◽  
Finite Linear ◽  
Complex Number Field

Let G be a finite subgroup of GLn(C) acting naturally on an affine space Cn of dimension n over the complex number field C and denote by Cn/G the quotient variety of Cn under this action of G. The purpose of this paper is to determine G completely such that Cn/G is a complete intersection (abbrev. CI.) i.e. its coordinate ring is a C.I. when n > 10. Our main result is (5.1). Since the subgroup N generated by all pseudo-reflections in G is a normal subgroup of G and Cn/G is obtained as the quotient variety of without loss of generality, we may assume that G is a subgroup of SLn(C) (cf. [6, 16, 24, 25]).

scholarly journals Normal subgroups of finite multiply transitive permutation groups

1974 ◽  
Vol 53 ◽  
pp. 103-107 ◽  
Author(s):  
Eiichi Bannai
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Normal Subgroups ◽  
Multiply Transitive

Wagner [5] and Ito [2] proved the following theorems respectively.THEOREM OF WAGNER. Let G be a triply transitive permutation group on a set Ω = {1,2, …, n}, and let n be odd and n > 4. If H is a normal subgroup (≠1) of G, then H is also triply transitive on Ω.

Unipotent normal subgroups of skew linear groups

1989 ◽  
Vol 105 (1) ◽  
pp. 37-51 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Normal Subgroup ◽  
Matrix Group ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Matrix Groups ◽  
Division Rings ◽  
Additional Assumptions

A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one, a situation where unipotent elements present few problems!

scholarly journals Maximal normal subgroups of the integral linear group of countable degree

1976 ◽  
Vol 15 (3) ◽  
pp. 439-451 ◽  
Author(s):  
R.G. Burns ◽  
I.H. Farouqi
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Normal Subgroup ◽  
Vector Space ◽  
Free Abelian Group ◽  
Normal Structure ◽  
Normal Subgroups ◽  
Finite Degree ◽  
Infinite Rank ◽  
Countably Infinite

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

A NOTE ON SOLUBLE GROUPS WITH THE MINIMAL CONDITION ON NORMAL SUBGROUPS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350134 ◽  
Author(s):  
M. DE FALCO ◽  
F. DE GIOVANNI ◽  
C. MUSELLA
Keyword(s):  
Finite Rank ◽  
Soluble Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Infinite Rank

It is known that there exist soluble groups of infinite rank which satisfy the minimal condition on normal subgroups. We prove here that if G is any soluble group satisfying the minimal condition on normal subgroups of infinite rank, then either G has finite rank or it satisfies the minimal condition on normal subgroups.

On Local Nilpotency of the Normal Subgroups of a Group

2016 ◽  
Vol 23 (03) ◽  
pp. 531-540
Author(s):  
Zhirang Zhang ◽  
Jiachao Li
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Maximal Subgroups ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Proper Subclass ◽  
Locally Nilpotent ◽  
Satisfy Property

A group G is said to have property μ whenever N is a non-locally nilpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property ν satisfy property μ, where a group G is said to have property ν if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP(G) is the Hirsch-Plotkin radical of G, and Φf(G) is the intersection of all the maximal subgroups of finite index in G (here Φf(G)=G if no such maximal subgroups exist). It is shown that a group G has property μ if and only if HP(G/Φf(G))=HP(G)/Φf(G) and that the class of groups with property ν is a proper subclass of that of groups with property μ. Also, the structure of the normal subgroups of a group: nilpotency with the minimal condition, is investigated.

scholarly journals Normal subgroups in the group of column-finite infinite matrices

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Waldemar Hołubowski ◽  
Martyna Maciaszczyk ◽  
Sebastian Zurek
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Division Ring ◽  
Classical Result ◽  
Normal Subgroups ◽  
Linear Transformations ◽  
Finite Dimensional ◽  
Infinite Cardinality ◽  
Positive Integers ◽  
Dimensional Range

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

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