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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24 CHAPTER 3<br />
since γ3(G) ⊆ (G ′ ) p , we have that k ≡ 1 (mod p). Therefore, aC has<br />
p-power order and G/C is a p-group.<br />
Since we have already closed <strong>the</strong> nilpotent case, by Lemma 3.2.1<br />
we may assume that G/C has order 2 m p r with m > 0, r ≥ 0, or it is<br />
divisible by some odd prime not equal to p. These two cases will be<br />
discussed in <strong>the</strong> sequel.<br />
Choose <strong>the</strong> element 〈aC〉 and <strong>the</strong> integer k such that G/C = 〈aC〉<br />
and x k = x a . We claim that<br />
(3.6) [(x − 1) 2l<br />
, a] ≡ (k 2l<br />
− 1)a(x − 1) 2l<br />
for any l ≥ 0. Using <strong>the</strong> well-known equation<br />
(3.7) (x s − 1) =<br />
we have<br />
as we claimed.<br />
[(x − 1) 2l<br />
, a] ≡ a<br />
k <br />
i=1<br />
s<br />
i=1<br />
s<br />
i<br />
(x − 1) ,<br />
i<br />
(mod I(G ′ ) 2l +1 ),<br />
2l k<br />
i<br />
(x − 1) −(x − 1) i<br />
2l<br />
<br />
≡ a k 2l<br />
(x − 1) 2l<br />
− (x − 1) 2l<br />
≡ (k 2l<br />
− 1)a(x − 1) 2l<br />
(mod I(G ′ ) 2l +1 ),<br />
Lemma 3.2.2. Let G be a group with cyclic commutator subgroup <strong>of</strong><br />
order p n , where p is an odd prime, and let char(F ) = p. If G/C has<br />
order 2 m p r <strong>the</strong>n<br />
[I(G ′ ) 2m i , I(G ′ ) 2 m j ] ⊆ I(G ′ ) 2 m i+2 m j+1 .<br />
Pro<strong>of</strong>. With <strong>the</strong> notation introduced above, <strong>the</strong> assumption ensures<br />
<strong>the</strong> congruence k2m ≡ 1 (mod p), thus we can apply (3.6) to conclude<br />
<strong>the</strong> inclusion<br />
[ω(F G ′ ) 2m<br />
, F G] ⊆ I(G ′ ) 2m +1<br />
.