Autocommutators and the autocommutator subgroup

2006 ◽  
pp. 137-146 ◽  
Author(s):  
David Garrison ◽  
Luise-Charlotte Kappe ◽  
Denise Yull
Keyword(s):  
Autocommutator Subgroup

Some properties of left 2α-Engel subgroups

2020 ◽  
pp. 2150068
Author(s):  
Ali Mohammad Z. Mehrjerdi ◽  
Mohammad Reza R. Moghaddam ◽  
Mohammad Amin Rostamyari
Keyword(s):  
Upper Bound ◽  
Factor Group ◽  
Central Factor ◽  
Famous Result ◽  
Commutator Formula ◽  
Autocommutator Subgroup ◽  
Derived Subgroup ◽  
The Absolute

In 1904, Schur proved his famous result which says that if the central factor group of a given group is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In the present paper, for a given automorphism [Formula: see text] of the group [Formula: see text], we introduce the concept of left [Formula: see text]-Engel, [Formula: see text], and [Formula: see text]-Engel commutator, [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

On autocommutators and the autocommutator subgroup in infinite abelian groups

2018 ◽  
Vol 30 (4) ◽  
pp. 877-885
Author(s):  
Luise-Charlotte Kappe ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Automorphism Group ◽  
Abelian Groups ◽  
Torsion Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Minimal Order ◽  
Autocommutator Subgroup ◽  
Free Abelian Groups ◽  
Torsion Free

Abstract It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

Some results of left and right α-commutators of groups

2019 ◽  
pp. 2050148
Author(s):  
Marziyeh Haghparast ◽  
Mohammad Reza R. Moghaddam ◽  
Mohammad Amin Rostamyari
Keyword(s):  
Upper Bound ◽  
Factor Group ◽  
Central Factor ◽  
Famous Result ◽  
Commutator Formula ◽  
Autocommutator Subgroup ◽  
Derived Subgroup ◽  
Left And Right ◽  
The Absolute

In [Formula: see text], Schur proved his famous result which says that if the central factor group of a given group [Formula: see text] is finite, then so is its derived subgroup. In [Formula: see text], Hegarty showed that if the absolute central factor group, [Formula: see text], is finite, then so is its autocommutator subgroup, [Formula: see text]. In this paper, we introduce the concept of left and right [Formula: see text]-commutator, [Formula: see text], and [Formula: see text], where [Formula: see text] is an automorphism of the group [Formula: see text]. Then under some condition, we prove that the finiteness of [Formula: see text] implies that [Formula: see text] is also finite. We also construct an upper bound for the order of [Formula: see text] in terms of the order of [Formula: see text].

Some properties of central kernel and central autocommutator subgroups

2016 ◽  
Vol 15 (07) ◽  
pp. 1650128 ◽  
Author(s):  
Samaneh Davoudirad ◽  
Mohammad Reza R. Moghaddam ◽  
Mohammad Amin Rostamyari
Keyword(s):  
Classical Result ◽  
Commutator Subgroup ◽  
Upper Bounds ◽  
Autocommutator Subgroup ◽  
Central Quotient

A classical result of Schur states that if the central quotient [Formula: see text] of a group [Formula: see text] is finite, then the commutator subgroup [Formula: see text] is also finite. In this paper we introduce the notion of central autocommutator subgroup of a given group [Formula: see text]. We study this concept and give some new results concerning the central kernel subgroup of [Formula: see text], which was first introduced by F. Haimo in 1955. More precisely, the analogue of Schur’s result is proved. We also construct some upper bounds for the order of central kernel and central autocommutator subgroups of [Formula: see text] in terms of the order of central kernel quotient of [Formula: see text].

scholarly journals On the autocommutator subgroup and absolute centre of a group

2011 ◽  
Vol 341 (1) ◽  
pp. 150-157 ◽  
Author(s):  
Heiko Dietrich ◽  
Primož Moravec
Keyword(s):  
Autocommutator Subgroup ◽  
Absolute Centre

Some properties of central kernel quotient of a group

2020 ◽  
pp. 2150047
Author(s):  
Samaneh Davoudirad ◽  
Mohammad Reza R. Moghaddam ◽  
Mohammad Amin Rostamyari
Keyword(s):  
Nilpotent Groups ◽  
Central Automorphism ◽  
Autocommutator Subgroup

The authors, in 2016, introduced the notion of central autocommutator subgroup of a group [Formula: see text] and proved some new results concerning the central kernel subgroup of [Formula: see text], which was introduced earlier by Haimo in [Formula: see text]. In this paper, we establish some results on central kernel quotient of [Formula: see text]. Finally, we introduce the concept of central autonilpotent (henceforth [Formula: see text]-nilpotent) groups and determine its relationship with the nilpotency of the subgroups of central automorphism of [Formula: see text].

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