On the Derived Length of Lie Solvable Group Algebras

36 CHAPTER 3
3.2.3. Finally, let m > 1. Since ω s(m)
d−1(F H) = 0, Lemma 3.2.5 forces
δ [d] (F H) = 0. Hence d + 1 ≤ dlL(F H) ≤ dlL(F G), which completes
the proof.
3.3 Summarized result
The determination of the values of dlL(F G) and dl L (F G) for the
case when G ′ is cyclic of odd order is a consequence of Theorems 3.1.2
and 3.2.7.
Theorem 3.3.1. Let G be a group with cyclic commutator subgroup of
order p n , where p is an odd prime, and let F be a field of characteristic
p.
(i) If G/CG(G ′ ) has order p r (that is G is nilpotent), then
dlL(F G) = dl L (F G) = ⌈log 2(p n + 1)⌉.
(ii) If the order of G/CG(G ′ ) is divisible by some odd prime q = p,
then
dlL(F G) = dl L (F G) = ⌈log 2(2p n )⌉.
(iii) If G/CG(G ′ ) has order 2 m p r with m > 0, then
dlL(F G) = dl L (F G) = d + 1,
where d is the minimal integer for which s (m)
d
Remarks on the theorem
≥ pn holds.
When (iii) of Theorem 3.3.1 holds for the group G we are able
to determine explicitly the values of dlL(F G) and dl L (F G) in the
following cases:
(i) We claim that if G/C has order 2p r , then
dlL(F G) = dl L (F G) = ⌈log 2(3p n /2)⌉.