Group algebra whose unit group is locally nilpotent Victor Bovdi - - PowerPoint PPT Presentation
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
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) = { ∑
g∈G
αgg ∈ U(FG) | ∑
g∈G
αg = 1}. 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.
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.
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.
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:
In Proc. Internat. Congr. Math. (Moscow, 1966), pages 284–289. Izdat. „Mir”, Moscow, 1968. For an overview we recommend the survey paper:
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 (x1, x2) = x−1
1 x−1 2 x1x2
and (x1, . . . , xn) = ( (x1, . . . , xn−1), xn ) , (x1, . . . , xn ∈ 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
condition. Israel J. Math., 67(3):287–290, 1989.
multiplicative group of a group algebra
multiplicative group of a group algebra
Victor Bovdi Group algebra whose unit group is locally nilpotent
The non-modular case was solved by A. Bovdi and I. Khripta:
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 = 2t − 1, the exponent of t(G) divides p2 − 1 and gag−1 = ap for all a ∈ t(G) and g ∈ 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):
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):
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
See Theorems 1.2 and 1.3 in
See Theorem 1.3 in
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 = Sylp(G) × A, where Sylp(G) ̸= ⟨1⟩, G′ ≤ Sylp(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
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 = 2t − 1, the exponent of t(G) divides p2 − 1 and g−1ag = ap for all a ∈ t(G) and g ∈ G \ CG(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.
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)
), 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:
. 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.
(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
In the paper:
locally nilpotent, 1–5, Januar, 2017. (submitted for publication) we characterized those modular group algebras FG whose groups of the unitary units V∗ are locally nilpotent. Note that there exist group algebras FG whose group of units V(FG) are not locally nilpotent, but their groups of the unitary units V∗ are locally nilpotent.
Victor Bovdi Group algebra whose unit group is locally nilpotent
The following trivial consequence of our Theorem gives a generalization of a result of G.T. Lee, S. Sehgal and E. Spinelli which was obtained for the case of characteristic of F different from 2.
algebra of a group G over the field F of characteristic 2. The following conditions are equivalent: (i) V(FG) is nilpotent; (ii) V∗ is nilpotent; (iii) G is nilpotent and G′ is a finite 2-group.
Victor Bovdi Group algebra whose unit group is locally nilpotent
Victor Bovdi Group algebra whose unit group is locally nilpotent