Modular group algebras with maximal Lie nilpotency indices

Abstract

In the present paper we give the full description of the Lie nilpotent group algebras which have maximal Lie nilpotency indices.

Full text

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arXiv:math/0507248v1 [math.RA] 12 Jul 2005
Publ. Math. Debrecen
Manuscript (.....)
Modular group algebras with
maximal Lie nilpotency indices
By VICTOR BOVDI (Debrecen) and ERNESTO SPINELLI (Lecce)
Dedicated to the memory of Professor Jen˝ o Erd˝ os
Abstract. In the present paper we give the full description of the Lie nilpo-
tent group algebras which have maximal Lie nilpotency indices.
1. Introduction
Let R be an associative algebra with identity. The algebra R can be
regarded as a Lie algebra, called the associated Lie algebra of R, via the
Lie commutator [x,y] = xy − yx, for every x,y ∈ R. Set
[x1,...,xn−1,xn] = [[x1,...,xn−1],xn],
where x1,...,xn∈ R. The n-th lower Lie power R[n]of R is the associative
ideal generated by all Lie commutators [x1,...,xn], where R[1]= R and
x1,...,xn∈ R. By induction, we define the n-th upper Lie power R(n)of
R as the associative ideal generated by all Lie commutators [x,y], where
R(1)= R and x ∈ R(n−1), y ∈ R.
An algebra R is called Lie nilpotent if there exists m such that R[m]=
0. The minimal integers m,n such that R[m]= 0 and R(n)= 0 are called
Mathematics Subject Classification: 16S34, 17B30.
Key words and phrases: Group algebras, Lie nilpotency indices.
The research was supported by OTKA No.T 037202 and No.T 038059.

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2VICTOR BOVDI (Debrecen) and ERNESTO SPINELLI (Lecce)
the Lie nilpotency index and
denoted they are by tL(R) and tL(R), respectively.
An algebra R is called Lie hypercentral if for any sequence {ai} of
elements of R there exists some n, such that [a1,...,an] = 0.
Let KG be the group algebra of a group G over a field K of character-
istic char(K) = p > 0. According to [2, 9] for the noncommutative group
algebra KG the following statements are equivalent: (a) KG is Lie nilpo-
tent (b) KG is Lie hypercentral; (c) char(K) = p > 0, G is nilpotent and
its commutator subgroup G′is a finite p-group. It is well known [8, 13],
if KG is Lie nilpotent then tL(KG) ≤ tL(KG) ≤ |G′| + 1. Moreover, ac-
cording to [1], if char(K) > 3, then tL(KG) = tL(KG). But the question
when tL(KG) = tL(KG) for char(K) = 2,3 is still open.
In the present paper we investigate the group algebras KG for which
tL(KG) is maximal, i.e.tL(KG) = |G′| + 1.
finite p-group and char(K) ≥ 5, then as Shalev [12] proved that tL(KG)
is maximal if and only if G′is cyclic. We give a complete characterization
by proving the following:
the upper Lie nilpotency index of R and
In particular, if G is a
Theorem 1. Let KG be a Lie nilpotent group algebra with
char(K) = p > 0. ThentL(KG) = |G′| + 1
following conditions holds:
(1) G′is cyclic;
(2) p = 2 and G′is the noncyclic of order 4 and γ3(G) ?= 1.
if and only if one of the
Corollary 1. Let KG be a Lie nilpotent group algebra with
char(K) = p > 0. IftL(KG) = |G′| + 1,thentL(KG) = tL(KG).
By Du’s Theorem [4], the previous result lists also the group algebras
KG whose group of units U(KG) has maximal nilpotency class under the
assumption that G is a finite p-group. Note that Konovalov [7] proved
that U(KG) of 2-groups of maximal class over field K with char(K) = 2
have the maximal nilpotency class.
We use the standard notation for a group G:
tini subgroup of G;gh:= h−1gh
γi(G) means the i-th term of the lower central series of G, i.e.
Φ(G) denotes the Frat-
and (g,h) := g−1h−1gh,(g,h ∈ G);
γ1(G) = G,γi+1(G) =?γi(G),G?
(i ≥ 1).

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Modular group algebras with maximal Lie nilpotency indices3
Moreover, Cnis the cyclic group of order n and set
Q2n
=? a,b | a2n−1= 1,
? a,b | a2n−1= b2= 1,
? a,b | a2n−1= b2= 1,
? a,b | a2n−1= b2= 1,
b2= a2n−2,
ab= a−1?,
ab= a−1+2n−2?,
ab= a1+2n−2?,
ab= a−1?,withn ≥ 3;
D2n
=withn ≥ 3;
SD2n
= withn ≥ 4;
MD2n
=
withn ≥ 4.
2. Preliminaries
Let K be a field of characteristic p > 0 and G a group. We consider a
sequence of subgroups of G setting
D(m)(G) = G ∩ (1 + KG(m)),(m ≥ 1).
The subgroup D(m)(G) is called the m-th Lie dimension subgroup of KG.
It is possible to describe the D(m)(G)’s in terms of the lower central series
of G in the following manner ( [8], p.44)
D(m+1)(G) =





G
G′
?D(m)(G),G?(D(⌈m
if
if
if
m = 0;
m = 1;
m ≥ 2.
p⌉+1)(G))p
(1)
where ⌈m
Put pd(m):= [D(m)(G) : D(m+1)(G)]. If KG is Lie nilpotent, according
to Jennings’ theory [10] for the Lie dimension subgroups, we get
p⌉ is the smallest integer greater thanm
p.
tL(KG) = 2 + (p − 1)
?
m≥1
md(m+1).
Lemma 1. ([11, 12]) Let K be a field with char(K) = p > 0 and G a
nilpotent group such that G′is a finite p-group with exp(G′) = pl.
(1) If d(m+1)= 0 and m is a power of p, then D(m+1)(G) = 1.
(2) If d(m+1)= 0 and pl−1divides m, then D(m+1)(G) = 1.
Lemma 2. Let p,s,n ∈ N and m0,...,ms−1the non-negative integers
such that s < n and?s−1
i=0mi= n. Then
?s−1
i=0mipi<?n−1
i=0pi.

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4VICTOR BOVDI (Debrecen) and ERNESTO SPINELLI (Lecce)
Proof. By the assumptions, there exist integers 0 ≤ j1< ··· < jk≤
s−1 such that mjl> 1 for every 1 ≤ l ≤ k. Since ps> pjk, we obtain that
s−1
?
i=0
mipi=
s−1
?
i=0
pi+
k
?
i=1
s−1
?
i=0
(mji− 1)pji
≤pi+ pjk(n − s) <
s−1
?
i=0
pi+
n−1
?
i=s
pi.
?
Lemma 3. Let K be a field with char(K) = p > 0 and G a nilpotent
group such that|G′| = pn. Then
d(pi+1)= 1 and d(j)= 0,where 0 ≤ i ≤ n − 1,
tL(KG) = |G′| + 1 if and only if
j ?= pi+ 1and j > 1.
Proof. If d(pi+1)= 1 for 0 ≤ i ≤ n − 1 and d(j)= 0 for j > 1, then
tL(KG) = 2 + (p − 1)
n−1
?
i=0
pi= 1 + pn= |G′| + 1.
In order to prove the other implication, we preliminarily remark that
?
m≥2
d(m)= n(2)
that is an immediate consequence of the definition of d(j)’s. Now we sup-
pose that there exists 0 ≤ j ≤ n − 1 such that d(pj+1)= 0. Let s be the
minimal integer for which d(ps+1)= 0. From (1) it follows at once that
s ?= 0 and by (1) of Lemma 1 we have that D(ps+1)(G) = 1 and so d(r)= 0
for every r ≥ ps+ 1. It is immediate by (2) that α :=?s−1
Let us consider the following two cases: α = n and α < n. If α = n then,
according to Lemma 2, we have that
i=0d(pi+1)≤ n.
tL(KG) = 2 + (p − 1)
s−1
?
i=0
pid(pi+1)< 2 + (p − 1)
n−1
?
i=0
pi= |G′| + 1.
If α < n by (2) there exists at least one j > 1 such that d(j)?= 0 and
j ?= pi+1. Suppose d(j1),...,d(jk)are all of such d(j)’s, where j1< ··· < jk.

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Modular group algebras with maximal Lie nilpotency indices5
Clearly, jk≤ ps. According to Lemma 2 for the case α > s, we obtain
tL(KG) = 2 + (p − 1)
s−1
?
i=0
α−1
?
i=0
α−1
?
i=0
α−1
?
i=0
pid(pi+1)+ (p − 1)
k
?
i=1
(ji− 1)d(ji)
≤ 2 + (p − 1)pi+ (p − 1)(jk− 1)(n − α)
< 2 + (p − 1)
pi+ (p − 1)ps(n − α)
< 2 + (p − 1)
pi+ (p − 1)
n−1
?
i=α
pi= |G′| + 1.
So, if tL(KG) is maximal, then d(pj+1)> 0 for each 0 ≤ j ≤ n−1 and, by
(2), the Lemma is proved.
?
Corollary 2. Let K be a field with char(K) = p > 0 and G a nilpotent
group with |G′| = pn. IftL(KG) = |G′| + 1, then |D(pi+1)(G)| = pn−i,
for 0 ≤ i ≤ n.
Proof. By Lemma 3, it is easy to check that
D(p0+1)(G) ⊃ D(p1+1)(G) ⊃ D(p1+2)(G) = ···
··· = D(pi+1)(G) ⊃ D(pi+2)(G) = ···
= D(ps+1)(G) ⊃ D(ps+2)(G) = 1
for some s ∈ N. Clearly, |D(ps+1)(G)| = p,
|D(pi+1)(G)| = ps−i+1, so |D(p0+1)(G)| = ps+1= pnand s = n − 1.
|D(ps−1+1)(G)| = p2and
?
Lemma 4. Let K be a field with char(K) = p > 0 and G a nilpotent
group with G′a finite p-group such that tL(KG) = |G′| + 1.
(1) If p > 2 then G′is cyclic.
(2) If p = 2 then G′has at most two generators.
Proof. Assume that |G′| = pn. Let us prove that
|Φ(G′)| ≥
?pn−1
2n−2
if
if
p ?= 2;
p = 2.