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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2.2 THE GENERALIZED RESULT 17<br />
As <strong>the</strong> following examples show, Theorem 2.2.2 breaks down without<br />
<strong>the</strong> condition γ3(G) ⊆ (G ′ ) p .<br />
• Let G be a group with G ′ = C2 × C2 such that γ3(G) = 1 and<br />
let char(F ) = 2. Then γ3(G) (G ′ ) 2 and, by Proposition 2.1.2,<br />
dl L (F G) > 2. So dl L (F G) = ⌈log 2(tN(G ′ ) + 1)⌉, because now<br />
⌈log 2(tN(G ′ ) + 1)⌉ = 2.<br />
• Let G be a group with G ′ = C3×C3×C3 such that γ3(G) = 1 and<br />
let char(F ) = 3. Then γ3(G) (G ′ ) 3 and, by Proposition 2.1.3,<br />
dl L (F G) > 3. It follows that dl L (F G) = ⌈log 2(tN(G ′ ) + 1)⌉,<br />
because ⌈log 2(tN(G ′ ) + 1)⌉ = 3.
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