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.

A note on Lie nilpotent group rings

1992 ◽  
Vol 45 (3) ◽  
pp. 503-506 ◽  
Author(s):  
R.K. Sharma ◽  
Vikas Bist
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Group Rings ◽  
Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

GROUP ALGEBRAS WITH UNIT GROUPS OF DERIVED LENGTH THREE

2010 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Characteristic P ◽  
Derived Length ◽  
A Finite Group ◽  
Necessary And Sufficient

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained.

On the Lower Bound of the Derived Length of the Unit Group of a Nontorsion Group Algebra

2019 ◽  
Vol 23 (2) ◽  
pp. 457-466
Author(s):  
Tibor Juhász ◽  
Gregory T. Lee ◽  
Sudarshan K. Sehgal ◽  
Ernesto Spinelli
Keyword(s):  
Lower Bound ◽  
Group Algebra ◽  
Unit Group ◽  
Derived Length

ON THE DERIVED LENGTH OF THE GROUP OF UNITS OF A GROUP ALGEBRA

2007 ◽  
Vol 06 (06) ◽  
pp. 991-999 ◽  
Author(s):  
ZSOLT BALOGH ◽  
YUANLIN LI
Keyword(s):  
Group Algebra ◽  
Characteristic P ◽  
Derived Length ◽  
Group Of Units ◽  
Derived Subgroup

Let G be a group with cyclic derived subgroup of p-power order (p > 2), and F a field of characteristic p. We determine the derived length of the group of units of the group algebra FG.

A note on Lie nilpotent group algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950163
Author(s):  
Meena Sahai ◽  
Bhagwat Sharan
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Index Formula ◽  
Group Algebras ◽  
Arbitrary Group ◽  
Nilpotency Index

Let [Formula: see text] be an arbitrary group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index [Formula: see text] of the group algebra [Formula: see text]. We also give improved bounds for [Formula: see text], where [Formula: see text] is the number of independent generators of the finite abelian group [Formula: see text]. Furthermore, we give a description of the Lie nilpotent group algebra [Formula: see text] with [Formula: see text] or [Formula: see text]. We also show that for [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the upper Lie nilpotency index of [Formula: see text].

On the derived length of the unit group of a group algebra

2010 ◽  
Vol 13 (4) ◽  
Author(s):  
Francesco Catino ◽  
Ernesto Spinelli
Keyword(s):  
Group Algebra ◽  
Unit Group ◽  
Derived Length

scholarly journals Group algebras with an Engel group of units

2006 ◽  
Vol 80 (2) ◽  
pp. 173-178 ◽  
Author(s):  
A. Bovdi
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Commutator Subgroup ◽  
Factor Group ◽  
Group Algebras ◽  
Engel Group ◽  
Group Of Units

AbstractLet F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350044
Author(s):  
TIBOR JUHÁSZ ◽  
ENIKŐ TÓTH
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Group Algebras ◽  
Augmentation Ideal ◽  
Characteristic P ◽  
Derived Length ◽  
Commutator Identity ◽  
Nilpotency Index ◽  
Powerful Group

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

Group Algebras with Minimal Strong Lie Derived Length

2008 ◽  
Vol 51 (2) ◽  
pp. 291-297 ◽  
Author(s):  
Ernesto Spinelli
Keyword(s):  
Solvable Group ◽  
Group Algebra ◽  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Derived Length ◽  
Necessary And Sufficient

AbstractLet KG be a non-commutative strongly Lie solvable group algebra of a group G over a field K of positive characteristic p. In this note we state necessary and sufficient conditions so that the strong Lie derived length of KG assumes its minimal value, namely [log2(p + 1)].

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