On the Derived Length of Lie Solvable Group Algebras

6.1 PRELIMINARY RESULTS 61
Proposition 6.1.1 (V. Bódi and E. Spinelli [13]). Let G be a nilpotent
group with commutator subgroup of order p n and let
char(F ) = p. Then tL(F G) = p n + 1 if and only if one of the following
conditions holds:
(i) G ′ is cyclic;
(ii) p = 2 and G ′ is the noncyclic of order 4 and γ3(G) = 1.
Assume that G ′ has order p n and t L (F G) < p n +1. Then, according
to (6.3), the next highest possible value of t L (F G) is p n − p + 2. If
t L (F G) = p n − p + 2 then we shall say that F G has almost maximal
upper Lie nilpotency index. Our goal is to determine these group
algebras.
To this we need the following results:
Proposition 6.1.2 (A.K. Bhandari and I.B.S. Passi [2]). Let F G be
a Lie nilpotent group algebra of characteristic p > 3. Then tL(F G) =
t L (F G).
Proposition 6.1.3 (Sahlev [26, 28]). Let G be a nilpotent group whose
commutator subgroup has order p n and exponent p l and let char(F ) =
p.
(i) If d(m+1) = 0 and m is a power of p, then D(m+1)(G) = 〈1〉.
(ii) If d(m+1) = 0 and p l−1 divides m, then D(m+1)(G) = 〈1〉.
(iii) If p ≥ 5 and tL(F G) < p n + 1, then tL(F G) ≤ p n−1 + 2p − 1.
Proposition 6.1.4 (V. Bódi and E. Spinelli [13]). Let G be a nilpotent
group with commutator subgroup of order p n and let
char(F ) = p. Then t L (F G) = p n + 1 if and only if d (p i +1) = 1 and
d(j) = 0, where 0 ≤ i ≤ n − 1, j = p i + 1 and j > 1.
Let P be a finite abelian p-group, P = 〈a1〉 × 〈a2〉 × · · · × 〈as〉,
where ai is of order pmi and m1 ≥ m2 ≥ · · · ≥ ms. We call {ai} a
basis of P . Any g ∈ P can be written uniquely as g = a k1
1 a k2
2 · · · aks s
with 0 ≤ k < pmi . We will denote ki by g(i), or by g(ai) if there are
more bases of P considered.