On the Derived Length of Lie Solvable Group Algebras

46 CHAPTER 4
The following question is still open: under which conditions are the
Lie derived length and the strong Lie derived length of group algebras
equal? From our results a partial answer follows to this question.
Corollary 4.2.2. Let G be a group with cyclic commutator subgroup
of order p n , and let F be a field of characteristic p. Then dlL(F G) =
dl L (F G) if and only if one of the following conditions is satisfied:
(i) p is odd;
(ii) p = 2 and n ≤ 2;
(iii) p = 2, n ≥ 3 and the nilpotency class of G is at most n.
Proof. The statement follows immediately from Theorems 3.3.1 and
4.2.1
In the case when p = 2 and neither (ii) nor (iii) is satisfied, by
Theorem 2.2.2 we know that dl L (F G) = n + 1, but the exact value
of dlL(F G) is still unknown. To determine it the results of the next
chapter can be useful.
According to Proposition 2.1.1, if F G is strongly Lie solvable of
characteristic p > 0 and G is nonabelian, then dl L (F G) is at least
⌈log 2(p + 1)⌉. Now, we describe the group algebras of minimal strong
Lie derived length.
Corollary 4.2.3. Let F G be a strongly Lie solvable group algebra of
characteristic p > 0. Then dl L (F G) = ⌈log 2(p + 1)⌉ if and only if one
of the following conditions holds:
(i) p = 2 and G ′ is central elementary abelian subgroup of order 4;
(ii) G ′ is of order p and
a) either G ′ is central;
b) or G/CG(G ′ ) has order 2mpr with m > 0, r ≥ 0, and the
minimal integer d such that s (m)
d ≥ p, satisfies the inequality
2d − 1 < p.