A characteristically simple group

1954 ◽  
Vol 50 (4) ◽  
pp. 641-642 ◽  
Author(s):  
D. H. McLain
Keyword(s):  
Simple Group ◽  
Finite Order ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Characteristic Subgroup ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Finite Set ◽  
Group A ◽  
Trivial Centre

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.

Locally finite minimal non-FC-groups

1989 ◽  
Vol 105 (3) ◽  
pp. 417-420 ◽  
Author(s):  
Mahmut Kuzucuoglu ◽  
Richard E. Phillips
Keyword(s):  
Simple Group ◽  
Proper Subgroup ◽  
Positive Answer ◽  
Locally Finite ◽  
Finite Set

We recall that a group G is an FC-group if for every x∈G the set of conjugates {xg|g∈G} is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC-group: such groups are called minimal non-FC-groups. Locally finite minimal non-FC-groups with (G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non-FC-group is either a simple group or a p-group for some prime p. Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the followingTheorem. There exists no simple, locally finite, minimal non-FC-group.

scholarly journals Normal subgroups of invertibles and of unitaries in a C*-algebra

2019 ◽  
Vol 2019 (756) ◽  
pp. 285-319
Author(s):  
Leonel Robert
Keyword(s):  
Simple Group ◽  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Connected Component ◽  
C Algebra ◽  
Inner Automorphisms ◽  
Extra Assumption ◽  
Group Modulo

AbstractWe investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a {\mathrm{C}^{*}}-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple \mathrm{C}^{*}-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital \mathrm{C}^{*}-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.

scholarly journals Transitive action on finite points of a full shift and a finitary Ryan’s theorem

2017 ◽  
Vol 39 (06) ◽  
pp. 1637-1667 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Turing Machines ◽  
Finite Support ◽  
Transitive Action ◽  
Group Invariant ◽  
Finite Set ◽  
Transitivity Property ◽  
Full Shift ◽  
Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s  $V$ .

scholarly journals Maximal quotient rings of prime group algebras. II Uniform right ideals

1977 ◽  
Vol 24 (3) ◽  
pp. 339-349 ◽  
Author(s):  
John Hannah
Keyword(s):  
Group Algebra ◽  
Quotient Ring ◽  
Group Algebras ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Residually Finite ◽  
Zero Characteristic ◽  
Quotient Rings

AbstractSuppose KG is a prime nonsingular group algebra with uniform right ideals. We show that G has no nontrivial locally finite normal subgroups. If G is soluble or residually finite, or if K has zero characteristic and G is linear, then the maximal right quotient ring of KG is simple Artinian.

scholarly journals INTEGRAL GROUP RING OF THE MATHIEU SIMPLE GROUP M24

2012 ◽  
Vol 11 (01) ◽  
pp. 1250016 ◽  
Author(s):  
VICTOR BOVDI ◽  
ALEXANDER KONOVALOV
Keyword(s):  
Simple Group ◽  
Group Ring ◽  
Unit Group ◽  
Positive Answer ◽  
Integral Group Ring ◽  
Integral Group ◽  
Sporadic Group ◽  
Prime Graphs ◽  
Zassenhaus Conjecture

We study the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M24. As a consequence, for this group we give a positive answer to the question by Kimmerle about prime graphs.

A note on minimal coverings of groups by subgroups

2001 ◽  
Vol 71 (2) ◽  
pp. 159-168 ◽  
Author(s):  
R. A. Bryce ◽  
L. Serena
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups ◽  
Finite Set ◽  
Group A ◽  
Abelian Subgroups

AbstractA cover for a group is a finite set of subgroups whose union is the whole group. A cover is minimal if its cardinality is minimal. Minimal covers of finite soluble groups are categorised; in particular all but at most one of their members are maximal subgroups. A characterisation is given of groups with minimal covers consisting of abelian subgroups.

scholarly journals Topologies on finite groups

1969 ◽  
Vol 1 (3) ◽  
pp. 315-317 ◽  
Author(s):  
Sidney A. Morris ◽  
H.B. Thompson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Discrete Topology ◽  
Normal Subgroups ◽  
Finite Set ◽  
A Finite Group

It has been shown by D. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 × 2n-2.We show that for admissable topologies on a finite group N ≦ 2n/r, where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound.

The derived length of the unit group of a group algebra — The case G′ = Sylp(G)

2016 ◽  
Vol 16 (08) ◽  
pp. 1750142 ◽  
Author(s):  
Tibor Juhász
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Derived Length ◽  
Group Of Units

Let [Formula: see text] be an odd prime, and let [Formula: see text] be a nilpotent group, whose commutator subgroup is finite abelian satisfying [Formula: see text] and [Formula: see text]. In this contribution, an upper bound is given on the derived length of the group of units of the group algebra of [Formula: see text] over a field of characteristic [Formula: see text]. Furthermore, we show that this bound is achieved, whenever [Formula: see text] is cyclic.

scholarly journals UNITS IN F2D2p

2013 ◽  
Vol 13 (02) ◽  
pp. 1350090 ◽  
Author(s):  
KULDEEP KAUR ◽  
MANJU KHAN
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Normal Subgroups ◽  
Canonical Involution ◽  
Unitary Subgroup ◽  
Bicyclic Units

Let p be an odd prime, D2p be the dihedral group of order 2p, and F2 be the finite field with two elements. If * denotes the canonical involution of the group algebra F2D2p, then bicyclic units are unitary units. In this note, we investigate the structure of the group [Formula: see text], generated by the bicyclic units of the group algebra F2D2p. Further, we obtain the structure of the unit group [Formula: see text] and the unitary subgroup [Formula: see text], and we prove that both [Formula: see text] and [Formula: see text] are normal subgroups of [Formula: see text].

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