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