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.

The Abelian Case of Solitar's Conjecture on Infinite Nielsen Transformations

1985 ◽  
Vol 28 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Olga Macedonska-Nosalska
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Infinite Rank ◽  
Abelian Case ◽  
Countably Infinite ◽  
Identical Transformation

AbstractThe paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.

scholarly journals On an infinite integral linear group

1971 ◽  
Vol 4 (3) ◽  
pp. 321-342 ◽  
Author(s):  
I.H. Farouqi ◽  
Hanna Neumann
Keyword(s):  
Normal Subgroup ◽  
Order Type ◽  
Normal Subgroups ◽  
Congruence Subgroups ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Infinite Integral ◽  
Countably Infinite ◽  
Dimensional Integral ◽  
The Continuum

This paper investigates the normal subgroup structure of the automorphism group Γ of a free abelian group A of countably infinite rank. The finitary automorphisms, that is those acting non-trivially only on a direct summand of A of finite rank, form a normal subgroup Φ of Γ; the sublattice of all normal subgroups of Γ contained in Φ is in fact the sublattice of normal subgroups of Φ and has a quite transparent structure. By contrast there is a profusion of normal subgroups of Γ not contained in Φ. For example, the collection of certain types of these normal subgroups, defined as generalizations of the congruence subgroups of finite dimensional integral linear groups, if partially ordered by inclusion, can be shown to contain infinitely many chains of the order type of the continuum.

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.

A note on automorphism groups of symmetric cubic graphs

2020 ◽  
pp. 2250018
Author(s):  
Jicheng Ma
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Simple Group ◽  
Minimal Normal Subgroup ◽  
Automorphism Groups ◽  
Cubic Graphs ◽  
Normal Subgroups ◽  
Interesting Consequence ◽  
Cubic Graph

We study [Formula: see text]-arc-transitive cubic graph [Formula: see text], and give a characterization of minimal normal subgroups of the automorphism group. In particular, each [Formula: see text] with quasi-primitive automorphism group is characterized. An interesting consequence of this characterization states that a non-solvable minimal normal subgroup [Formula: see text] contains at most 2 copies of non-abelian simple group when it acts transitively on arcs, or contains at most 6 copies of non-abelian simple group when it acts regularly on vertices.

The group of almost automorphisms of the countable universal graph

1989 ◽  
Vol 105 (2) ◽  
pp. 223-236 ◽  
Author(s):  
J. K. Truss
Keyword(s):  
Abelian Group ◽  
Simple Group ◽  
Permutation Group ◽  
Random Graph ◽  
Free Abelian Group ◽  
Natural Extension ◽  
Transitive Permutation Group ◽  
Normal Subgroups ◽  
Lattice Of Subgroups

The group Aut Γ of automorphisms of Rado's universal graph Γ (otherwise known as the ‘random’ graph: see [1]) and the corresponding groups Aut Γc for C a set of ‘colours’ with 2 ≤ |C| ≤ ℵ0, were studied in [4]. It was shown that Aut Γc is a simple group, and the possible cycle types of its members were classified. A natural extension of Aut Γc to a highly transitive permutation group on the same set is obtained by considering the ‘almost automorphisms’ of Γ. It is the purpose of the present paper to answer similar questions about the resulting group AAut Γc. Namely we shall classify its normal subgroups and the cycle types of its members. The main result on normal subgroups is summed up in Corollary 2·9, which says that the non-trivial normal subgroups of AAut Γc form a lattice isomorphic to the lattice of subgroups of the free Abelian group of rank n where n = |C| – 1, and for cycle types it will be shown that those occurring in AAut Γc are precisely the same as in Aut Γc except for those which are the product of finitely many cycles.

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.

scholarly journals Collineation groups of translation planes of small dimension

1981 ◽  
Vol 4 (4) ◽  
pp. 711-724 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Vector Space ◽  
Translation Plane ◽  
Small Dimension ◽  
Translation Planes ◽  
Normal Subgroups ◽  
Translation Complement ◽  
Collineation Groups

A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for “geometrically primitive”. If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup of orderw2a+bwherewadivides the dimension of the vector space. Similar conditions hold for solvable normal subgroups of geometrically primitive nonsolvable groups. When the dimension of the vector space is small there are restrictions on the group which might possibly be in the translation complement. We look at the situation for certain orders of the plane.

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