On the Derived Length of Lie Solvable Group Algebras

A POSSIBILITY OF APPLICATION 55
a) n = 2;
b) n = 3 and G is of class 4;
c) G has an abelian subgroup of index two.
Proof. Suppose first that p > 7. Then Proposition 5.1.1 states that
dlL(F G) ≥ ⌈log2(p + 1)⌉ ≥ 4. For odd p ≤ 7 the statement follows
directly from Theorem 3.3.1.
Let G ′ = 〈x | x 2n
= 1〉. The result follows from Theorem 3.3.1 for
n = 2 and n = 3. For n > 3, using induction on n, we shall show that if
dlL(F G) = 3 then C is abelian and G/C = 〈aC〉, where xa = x−1 (i.e.
G has an abelian subgroup of index two). Indeed, by Lemma 5.1.5,
this is true for n = 4. Let now n > 4 and dlL(F G) = 3 and assume
that the statement is true for every group with commutator subgroup
of order less than 2n . Set H = 〈x2n−1〉 ⊂ G ′
. Then dlL F (G/H) = 3
and (G/H) ′ = G ′ /H = 〈xH〉, and by inductive hypothesis we get
(xH) gH = x g H = x (−1)k
H
for all g ∈ G. It follows that x g = x i with
i ∈ {−1, 1, 2 n−1 − 1, 2 n−1 + 1},
i.e. exp(G/C) ≤ 2 and the statement follows from Lemma 5.1.4.
5.2 Group algebras of 2-groups of order 2 m
and exponent 2 m−2
The full description of the finite nonabelian 2-group of order 2 m
and exponent 2 m−2 can be found in [20]. These groups are:
• m ≥ 4
G1 = 〈a, b | a 2m−2
G2 = Q 2 m−1 × C2;
G3 = D 2 m−1 × C2;
= 1, b4 = 1, ab = a1+2m−3 〉;
G4 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = ca, bc = a2m−3 b〉;
G5 = 〈a, b, c | a2m−2 = 1, b2 = 1, c2 = 1, ab = ba, ac = ab, bc = cb〉;