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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GROUP ALGEBRAS WITH ALMOST MAXIMAL... 69<br />
Finally we can state <strong>the</strong> main <strong>the</strong>orem <strong>of</strong> this chapter.<br />
Theorem 6.2.4. Let F G be a <strong>Lie</strong> nilpotent group algebra over a field<br />
F <strong>of</strong> positive characteristic p. Then F G has upper almost maximal <strong>Lie</strong><br />
nilpotency index if and only if one <strong>of</strong> <strong>the</strong> following conditions holds:<br />
(i) p = 2, G is <strong>of</strong> class 2 and G ′ is noncyclic <strong>of</strong> order 4;<br />
(ii) p = 2, G is <strong>of</strong> class 4, G ′ = C4 × C2 and γ3(G) = C2 × C2;<br />
(iii) p = 2, G is <strong>of</strong> class 4 and G ′ is elementary abelian <strong>of</strong> order 8;<br />
(iv) p = 3, G is <strong>of</strong> class 3 and G ′ is elementary abelian <strong>of</strong> order 9.<br />
Pro<strong>of</strong>. Assuming that F G has upper almost maximal <strong>Lie</strong> nilpotency<br />
index, <strong>the</strong> statement is a consequence <strong>of</strong> <strong>the</strong> previous lemmas. The<br />
converse is obvious by (6.3).<br />
We mention that during <strong>the</strong> preparation <strong>of</strong> this <strong>the</strong>sis V. Bódi<br />
[10] proved <strong>the</strong> following statement: F G has upper almost maximal<br />
<strong>Lie</strong> nilpotency index if and only if F G has lower almost maximal <strong>Lie</strong><br />
nilpotency index.