On the Derived Length of Lie Solvable Group Algebras

62 CHAPTER 6
Proposition 6.1.5 (A.A. Bódi and J. Kurdics [8]). Let P be a finite
abelian p-group presented as above and let H be a proper subgroup of
P such that the order of the factor group HP p /P p is p r with r > 0.
Then the following statements hold:
(i) the function
ν : H \ P p → {1, 2, . . . , s}, ν(h) = min{j | gcd(h(j), p) = 1}
takes r distinct values v1 < v2 < · · · < vr;
(ii) there exist b1, b2, . . . , br ∈ H such that ν(bi) = vi, bi(vj) = 0 for
j < i, bi(vi) = 1, and if
{1, 2, . . . , s} = {u1, u2, . . . , us−r, v1, v2, . . . , vr},
u1 < u2 < · · · < us−r, and
A = 〈au1〉 × 〈au2〉 × · · · × 〈aus−r〉,
called the weak complement of H in P relative to the basis {ai},
then
P/A = 〈b1A〉 × 〈b2A〉 × · · · × 〈brA〉;
(iii) weak complements of H in P , relative to any basis, are all isomorphic
to each other;
(iv) if G is a nilpotent group of class 3 such that G ′ is a finite abelian
p-group and the order of γ3(G)(G ′ ) p /(G ′ ) p is p r with r > 0 then
tL(F G) = t L (F G) = tN(G ′ ) + tN(G ′ /A),
where A is the weak complement of γ3(G) in G ′ .
6.2 Group algebras with almost maximal
upper Lie nilpotency index
The description of the group algebras with almost maximal upper
Lie nilpotency index will be based on the following lemmas.