On the Derived Length of Lie Solvable Group Algebras

4.2 THE DESCRIPTION AND SOME CONSEQUENCES 47
Proof. Assume that G ′ has order p n . Then, as it is well-known,
tN(G ′ ) ≥ 1 + n(p − 1). Hence, if p = 2 and n ≥ 3, then tN(G ′ ) ≥ 4
and applying Proposition 2.1.1 we have that
⌈log 2(p + 1)⌉ = 2 < 3 = ⌈log 2(4 + 1)⌉ ≤ dl L (F G).
Let now p > 2 and n ≥ 2. Since tN(G ′ ) ≥ 2p − 1 and there exists i
such that p < 2 i < 2p, we get as before that
⌈log 2(p + 1)⌉ < ⌈log 2(2p)⌉ ≤ dl L (F G).
Thus, we have just proved that if dl L (F G) = ⌈log 2(p + 1)⌉ then either
G ′ = C2 × C2 and p = 2 or G ′ has order p. For p = 2 the statement
follows from Proposition 2.1.2 and for odd p from Theorem 3.3.1.