On the Derived Length of Lie Solvable Group Algebras

60 CHAPTER 6
(see Theorem 2.8 of [21]):
⎧
⎪⎨ G if m = 0;
(6.1) D(m+1)(G) = G
⎪⎩
′
D(m)(G), G
if m = 1;
(D(⌈ m
p ⌉+1)(G)) p if m ≥ 2,
where ⌈ m
p
⌉ is the upper integer part of m
p .
By [21] there also exists an explicit expression for D(m+1)(G):
(6.2) D(m+1)(G) =
Evidently,
(j−1)p i ≥m
γj(G) pi
.
G = D(1)(G) ⊇ D(2)(G) ⊇ · · · ⊇ D(m)(G) ⊇ · · · .
It is easy to check that the factor group D(k)(G)/D(k+1)(G) is an
elementary abelian p-group for any k ≥ 1. Put
p d (k) = [D(k)(G) : D(k+1)(G)].
According to Jennings’ theory [25] for the Lie dimension subgroups,
we get
(6.3) t L (F G) = 2 + (p − 1)
md(m+1),
and
(6.4)
m≥1
d(m) = n.
m≥2
As we have already mentioned, if G ′ has order p n then
tL(F G) ≤ t L (F G) ≤ p n + 1.
The Lie nilpotent group algebras with maximal upper (and lower)
Lie nilpotency indices have been described: