Group algebra whose unit group is locally nilpotent Victor Bovdi Department of Math. Sciences UAE University Al-Ain United Arab Emirates e-mail: vbovdi@gmail.com Topics on Groups and their Representations in honor of Professor Lino Di Martino Palazzo Feltrinelli, Gargnano sul Garda, October 9th - 11th, 2017 Victor Bovdi Group algebra whose unit group is locally nilpotent
Let U ( FG ) be the group of units of the group algebra FG of a group G over a field F of characteristic char ( F ) = p ≥ 0. U ( FG ) = V ( FG ) × U ( F ); where ∑ ∑ V ( FG ) = { α g g ∈ U ( FG ) | α g = 1 } . g ∈ G g ∈ G The group of normalized units V ( FG ) of a modular group algebra FG has a complicated structure and was studied in several papers. For an overview we recommend the survey paper of A. Bovdi: The group of units of a group algebra of characteristic p . Publ. Math. Debrecen , 52(1-2):193–244, 1998. Victor Bovdi Group algebra whose unit group is locally nilpotent
An explicit list of groups G and rings K for which V ( KG ) are nilpotent was obtained by I. Khripta. For the modular case in: The nilpotence of the multiplicative group of a group ring. Mat. Zametki , 11:191–200, 1972. For the non-modular case in : The nilpotence of the multiplicative group of a group ring. Latvian math. yearbook , Zinatne, Riga, 13:119–127, 1973. In the paper of A. Bovdi: Group algebras with a solvable group of units. Comm. Algebra , 33(10):3725–3738, 2005. it was completely determined when V ( FG ) is solvable. Victor Bovdi Group algebra whose unit group is locally nilpotent
It is well known that the Engel property of a group is close to its local nilpotency. A locally nilpotent group is always Engel! However these classes of groups do not coincide, for example see the famous E. Golod’s counterexample in: E. S. Golod. Some problems of Burnside type. In Proc. Internat. Congr. Math. (Moscow, 1966) , pages 284–289. Izdat. „Mir”, Moscow, 1968. For an overview we recommend the survey paper: G. Traustason. Engel groups. In Groups St Andrews 2009 in Bath. Vol. 2 , vol. 388 of London Math. Soc. Lecture Note Ser. , pp. 520–550. Cambridge Univ. Press, Cambridge, 2011. Victor Bovdi Group algebra whose unit group is locally nilpotent
A group G is said to be Engel if for any x , y ∈ G holds: ( x , y , y , . . . , y ) = 1 y is repeated sufficiently many times depending on x and y . We shall use the left-normed simple commutator notation ( x 1 , x 2 ) = x − 1 1 x − 1 2 x 1 x 2 and ( ) ( x 1 , . . . , x n ) = ( x 1 , . . . , x n − 1 ) , x n , ( x 1 , . . . , x n ∈ G ) . A group is called locally nilpotent if all its f. g. (finitely generated) subgroups are nilpotent. A locally nilpotent group is always Engel! Victor Bovdi Group algebra whose unit group is locally nilpotent
It is still a challenging problem whether V ( FG ) is an Engel group. This question has a long history: A. Shalev. On associative algebras satisfying the Engel condition. Israel J. Math. , 67(3):287–290, 1989. A. A. Bovdi and I. I. Khripta. The Engel property of the multiplicative group of a group algebra Dokl. Akad. Nauk SSSR , 314(1):18–20, 1990. A. A. Bovdi and I. I. Khripta. The Engel property of the multiplicative group of a group algebra Mat. Sb. , 182(1):130–144, 1991. A. A. Bovdi. Group algebras with an Engel group of units. J. Aust. Math. Soc. , 80(2):173–178, 2006. Victor Bovdi Group algebra whose unit group is locally nilpotent
The non-modular case was solved by A. Bovdi and I. Khripta: Theorem. Let FG be a non-modular group algebra. If U ( FG ) is an Engel group, then G is also an Engel group, the torsion part t ( G ) is an abelian group and one of the following conditions holds: (i) t ( G ) is central in G ; (ii) every idempotent in F [ t ( G )] is central in FG , F is a prime field of characteristic p = 2 t − 1, the exponent of t ( G ) divides p 2 − 1 and gag − 1 = a p for all a ∈ t ( G ) and g ∈ G outside of the centralizer of t ( G ) . Conversely, if G is an Engel group satisfying one of these conditions and G / t ( G ) is a u.p.-group, then U ( FG ) is an Engel group. Victor Bovdi Group algebra whose unit group is locally nilpotent
For the modular case there is no complete solution (A. Bovdi): Theorem. Let FG be a modular group algebra of characteristic p and assume that one of the following conditions hold: ◮ (i) G is solvable; ◮ (ii) p -Sylow subgroup P is solvable, normal in G , and contains a nontrivial finite subgroup N which is normal in G and p divides | N | ; ◮ (iii) p -Sylow subgroup P is finite. Then U ( FG ) is an Engel group if and only if G is locally nilpotent and G ′ is a p -group. As a matter of fact: In that case U ( FG ) is not only Engel but even locally nilpotent. Victor Bovdi Group algebra whose unit group is locally nilpotent
However, in modular case there is a full description of FG when V ( FG ) is a bounded Engel group (A. Bovdi): Theorem. Let FG be a modular group algebra of characteristic p . Then U ( FG ) is a bounded Engel group if and only if G is a nilpotent group with a normal subgroup H of p -power index such that H ′ is a finite p -group. In this case: FG is a bounded Engel algebra if and only if U ( FG ) is a bounded Engel group. Victor Bovdi Group algebra whose unit group is locally nilpotent
In several articles, M. Ramezan-Nassab attempted to describe the structure of groups G for which V ( FG ) are Engel (locally nilpotent) groups in the case when FG have only a finite number of nilpotent elements: see Theorem 1.5 in M. Ramezan-Nassab. Group algebras with locally nilpotent unit groups. Comm. Algebra , 44(2):604–612, 2016. See Theorems 1.2 and 1.3 in M. Ramezan-Nassab. Group algebras with Engel unit groups. J. Aust. Math. Soc. , 101(2):244–252, 2016. See Theorem 1.3 in M. Ramezan-Nassab. Group algebras whose p -elements form a subgroup. J. Algebra Appl. , 16(9):1750170, 7, 2017. Victor Bovdi Group algebra whose unit group is locally nilpotent
The following theorem gives a complete answer. Theorem 1. (V. Bovdi, 2017.) Let FG be the group algebra of a group G . If FG has only a finite number of non-zero nilpotent elements, then F is a finite field of char ( F ) = p . Additionally, if V ( FG ) is an Engel group, then V ( FG ) is nilpotent, G is a finite group such that G = Syl p ( G ) × A , where Syl p ( G ) ̸ = ⟨ 1 ⟩ , G ′ ≤ Syl p ( G ) and A is a central subgroup of G . The set of elements of finite orders of a group G (which is not necessarily a subgroup) is called the torsion part of G and is denoted by t ( G ) . Victor Bovdi Group algebra whose unit group is locally nilpotent
The next two theorems completely describe groups G with V ( FG ) locally nilpotent. Some special cases of the present Theorem were proved by I. Khripta and M. Ramezan-Nassab. Theorem 2. (V. Bovdi, 2017.) Let FG be a modular group algebra of a group G over the field F of positive characteristic p . The group V ( FG ) is locally nilpotent if and only if G is locally nilpotent and G ′ is a p -group. Theorem 3. (V. Bovdi, 2017.) Let FG be a non-modular group algebra of characteristic p ≥ 0. The group V ( FG ) is locally nilpotent if and only if G is a locally nilpotent group, t ( G ) is an abelian group and one of the following conditions holds: (i) t ( G ) is a central subgroup; (ii) F is a prime field of characteristic p = 2 t − 1, the exponent of t ( G ) divides p 2 − 1 and g − 1 ag = a p for all a ∈ t ( G ) and g ∈ G \ C G ( t ( G )) . Victor Bovdi Group algebra whose unit group is locally nilpotent
As a consequence of previous Theorems we obtain the classical result of I. Khripta. Corollary. (V. Bovdi, 2017.) Let FG be a modular group algebra of positive characteristic p . The group V ( FG ) is nilpotent if and only if G is nilpotent and G ′ is a finite p -group. The structure of V ( FG ) is the following: the group t ( G ) = P × D , where P is the p -Sylow subgroup of G , D is a central subgroup, 1 + I ( P ) is the p -Sylow subgroup of V ( FG ) and V ′ ≤ 1 + I ( G ′ ) . Moreover, if D is a finite abelian group, then we have the following isomorphism between abelian groups ) ∼ ( V ( FG ) / 1 + I ( P ) = V ( FD ) × ( G / t ( G ) × · · · × G / t ( G ) ) , � �� � n where n is the number of summands in the decomposition of FD into a direct sum of fields. Victor Bovdi Group algebra whose unit group is locally nilpotent
Let V ∗ ( KG ) be the unitary subgroup of the group V ( KG ) of normalized units of the group ring KG of a group G over the ring K , under the classical involution ∗ of KG . The group V ∗ ( KG ) has a complicated structure, has been actively studied and it has several applications. For instance, see the papers: S. P . Novikov. Algebraic construction and properties of Hermitian analogs of K -theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. I. II. Izv. Akad. Nauk SSSR Ser. Mat. , 34:253–288; ibid. 34 (1970), 475–500, 1970. J.-P . Serre. Bases normales autoduales et groupes unitaires en caractéristique 2 Transform. Groups , 19(2):643–698, 2014. Victor Bovdi Group algebra whose unit group is locally nilpotent
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