On the Derived Length of Lie Solvable Group Algebras
On the Derived Length of Lie Solvable Group Algebras
On the Derived Length of Lie Solvable Group Algebras
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10 CHAPTER 1<br />
1.2.3 <strong>Lie</strong> nilpotency indices <strong>of</strong> group algebras<br />
The object <strong>of</strong> <strong>the</strong> sixth chapter is <strong>the</strong> investigation <strong>of</strong> <strong>the</strong> <strong>Lie</strong> nilpotency<br />
indices <strong>of</strong> <strong>Lie</strong> nilpotent group algebras. For <strong>the</strong> noncommutative<br />
modular group algebra F G <strong>the</strong> next <strong>the</strong>orem from A.A. Bódi and I.I.<br />
Khripta [7] is well-known: The following statements are equivalent:<br />
(i) F G is <strong>Lie</strong> nilpotent; (ii) F G is upper <strong>Lie</strong> nilpotent; (iii) G is a<br />
nilpotent group whose commutator subgroup is a finite p-group and<br />
char(F ) = p.<br />
Since (F G) [n] ⊆ (F G) (n) for all n, tL(F G) ≤ t L (F G). Moreover,<br />
for characteristic p > 3 <strong>the</strong> equality is also guaranteed by a result<br />
<strong>of</strong> A.K. Bhandari and I.B.S. Passi [2], but <strong>the</strong> problem is still open<br />
o<strong>the</strong>rwise.<br />
According to [32], if F G is <strong>Lie</strong> nilpotent and G ′ has order p n , <strong>the</strong>n<br />
tL(F G) ≤ t L (F G) ≤ p n + 1.<br />
A. Shalev in [25] began to study <strong>the</strong> question when a <strong>Lie</strong> nilpotent<br />
group algebra has <strong>the</strong> maximal upper <strong>Lie</strong> nilpotency index, but <strong>the</strong><br />
complete description <strong>of</strong> such group algebras was given by V. Bódi<br />
and E. Spinelli in [13]. Joining this research we determine <strong>the</strong> group<br />
algebras whose upper <strong>Lie</strong> nilpotency index is ‘almost maximal’, that<br />
is, it takes <strong>the</strong> next highest possible value, namely p n − p + 2, where<br />
p n is <strong>the</strong> order <strong>of</strong> <strong>the</strong> commutator subgroup <strong>of</strong> <strong>the</strong> basic group.