scholarly journals Outer automorphisms of hypercentral p-groups

1995 ◽  
Vol 37 (2) ◽  
pp. 243-247
Author(s):  
Orazio Puglisi
Keyword(s):  
Nilpotent Group ◽  
Full Group ◽  
Finite Nilpotent Group ◽  
Group Of Automorphisms ◽  
Outer Automorphisms ◽  
Inner Automorphisms

In his celebrated paper [3] Gaschiitz proved that any finite non-cyclic p-group always admits non-inner automorphisms of order a power of p. In particular this implies that, if G is a finite nilpotent group of order bigger than 2, then Out (G) = Aut(G)/Inn(G) =≠1. Here, as usual, we denote by Aut (G) the full group of automorphisms of G while Inn (G) stands for the group of inner automorphisms, that is automorphisms induced by conjugation by elements of G. After Gaschiitz proved this result, the following question was raised: “if G is an infinite nilpotent group, is it always true that Out (G)≠1?”

A Class of Three-Generator, Three-Relation, Finite Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 36-40 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Finite Nilpotent Group ◽  
Image Position

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.The groups presented are defined as:with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.We prove the following result.THEOREM 1. Each of the groups presented is a finite soluble group.We state the following theorem proved by Macdonald (1).THEOREM 2. G1(α, β, 1) is a finite nilpotent group.1. In this section we make some elementary remarks.

Applications of Fibonacci sequences in a finite nilpotent group

2003 ◽  
Vol 141 (2-3) ◽  
pp. 565-578
Author(s):  
Engin Özkan ◽  
Hüseyin Aydın ◽  
Ramazan Dikici
Keyword(s):  
Nilpotent Group ◽  
Finite Nilpotent Group ◽  
Fibonacci Sequences

scholarly journals Finite nilpotent groups that coincide with their 2-closures in all of their faithful permutation representations

2018 ◽  
Vol 17 (04) ◽  
pp. 1850065
Author(s):  
Alireza Abdollahi ◽  
Majid Arezoomand
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
Finite Nilpotent Group ◽  
Closed Group ◽  
Odd Order ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

Let [Formula: see text] be any group and [Formula: see text] be a subgroup of [Formula: see text] for some set [Formula: see text]. The [Formula: see text]-closure of [Formula: see text] on [Formula: see text], denoted by [Formula: see text], is by definition, [Formula: see text] The group [Formula: see text] is called [Formula: see text]-closed on [Formula: see text] if [Formula: see text]. We say that a group [Formula: see text] is a totally[Formula: see text]-closed group if [Formula: see text] for any set [Formula: see text] such that [Formula: see text]. Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.

Small Directed Strongly Regular Graphs

2020 ◽  
Vol 27 (01) ◽  
pp. 11-30
Author(s):  
Štefan Gyürki
Keyword(s):  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Full Group ◽  
Group Of Automorphisms ◽  
Strongly Regular ◽  
Directed Strongly Regular Graphs

The goal of the present paper is to provide a gallery of small directed strongly regular graphs. For each graph of order n ≤ 12 and valency k < n/2, a diagram is depicted, its relation to other small directed strongly regular graphs is revealed, the full group of automorphisms is described, and some other nice properties are given. To each graph a list of interesting subgraphs is provided as well.

scholarly journals Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

scholarly journals THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003

2013 ◽  
Vol 23 (06) ◽  
pp. 1485-1496 ◽  
Author(s):  
V. S. ATABEKYAN
Keyword(s):  
Normal Subgroup ◽  
Free Groups ◽  
Group Of Automorphisms ◽  
Burnside Groups ◽  
Free Burnside Group ◽  
Trivial Center ◽  
Groups Of Automorphisms ◽  
Group B ◽  
Inner Automorphisms ◽  
Automorphism Tower

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

scholarly journals AXIOMATISABILITY OF THE CLASS OF MONOLITHIC GROUPS IN A VARIETY OF NILPOTENT GROUPS

2020 ◽  
Vol 102 (1) ◽  
pp. 67-76
Author(s):  
JOSHUA T. GRICE
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Subdirectly Irreducible ◽  
Finite Nilpotent Group ◽  
Finite Set

The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

scholarly journals NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP

2009 ◽  
Vol 52 (1) ◽  
pp. 169-177
Author(s):  
GÉRARD ENDIMIONI
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms ◽  
Free Soluble Group

AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

scholarly journals Arithmetic E8 Lattices with Maximal Galois Action

2009 ◽  
Vol 12 ◽  
pp. 144-165 ◽  
Author(s):  
Anthony Várilly-Alvarado ◽  
David Zywina
Keyword(s):  
Elliptic Curves ◽  
Weyl Group ◽  
Number Fields ◽  
Picard Group ◽  
Del Pezzo Surfaces ◽  
Full Group ◽  
Group Of Automorphisms ◽  
Intersection Pairing ◽  
Galois Action ◽  
Del Pezzo

AbstractWe construct explicit examples of E8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E8 and have maximal Galois action.Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.

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