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