Categorizing finite p-groups by the order of their non-abelian tensor squares

2016 ◽  
Vol 15 (05) ◽  
pp. 1650095 ◽  
Author(s):  
S. Hadi Jafari
Keyword(s):  
Upper Bound ◽  
Derived Subgroup

Let [Formula: see text] be a non-abelian [Formula: see text]-generator finite [Formula: see text]-group of order [Formula: see text]. Ellis and McDermott in 1996 proved that [Formula: see text]. In the present paper, we improve this upper bound and show that [Formula: see text]. Also the [Formula: see text]-groups with derived subgroup of order [Formula: see text] which attain the bound are obtained. Among other results, we classify all finite [Formula: see text]-groups of order [Formula: see text], for [Formula: see text], with [Formula: see text], where [Formula: see text].

scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

On the triple tensor product of prime-power groups

2020 ◽  
Vol 23 (5) ◽  
pp. 879-892
Author(s):  
S. Hadi Jafari ◽  
Halimeh Hadizadeh
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Upper Bound ◽  
Prime Power ◽  
Tensor Products ◽  
Lower Central Series ◽  
Central Series ◽  
Derived Subgroup

AbstractLet G be a finite p-group, and let {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of {\otimes^{3}G}, which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 1998, 2, 119–128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained. Furthermore, by improvement of a result about the exponent of {\otimes^{3}G} determined by G. Ellis [On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353 2001, 10, 4219–4234], we show that, when G is a nilpotent group of class at most 4, {\exp(\otimes^{3}G)} divides {\exp(G)}.

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].

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].

Sign in / Sign up

Export Citation Format

Share Document