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