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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3.2 THE BASIC GROUP IS NOT NILPOTENT 35<br />
Pro<strong>of</strong>. The upper bound ⌈log 2(2p n )⌉ on dl L (F G) is a consequence <strong>of</strong><br />
Proposition 2.1.1.<br />
The statement (i) follows directly from Lemma 3.2.6. In order to<br />
prove <strong>the</strong> inequality<br />
⌈log 2(3p n /2)⌉ ≤ dlL(F G) ≤ dl L (F G) ≤ ⌈log 2(2p n )⌉,<br />
it remains to show that if G/C has order 2 m p r , <strong>the</strong>n ⌈log 2(3p n /2)⌉ ≤<br />
dlL(F G). Since G is not nilpotent, Lemma 3.2.1 ensures m > 0, so<br />
<strong>the</strong>re is an element bC ∈ G/C <strong>of</strong> order 2. Let H = 〈x, b〉. Clearly,<br />
b 2 ∈ ζ(H) and x b = x −1 , <strong>the</strong>refore <strong>the</strong> factor group H = H/ζ(H) is<br />
isomorphic to <strong>the</strong> dihedral group <strong>of</strong> order 2p n , so by Lemma 3.2.3, if<br />
d is <strong>the</strong> minimal integer such that s (1)<br />
d ≥ pn <strong>the</strong>n we have<br />
d + 1 ≤ dlL(F H) ≤ dlL(F H) ≤ dlL(F G).<br />
At <strong>the</strong> same time, by (3.8) we have (2 d+2 − 1)/3 ≥ s (1)<br />
d ≥ pn , whence<br />
d + 1 ≥ ⌈log 2(3p n /2 + 1/2)⌉ follows. Since ⌈log 2(3p n /2 + 1/2)⌉ =<br />
⌈log 2(3p n /2)⌉, <strong>the</strong> <strong>the</strong> required inequality is guaranteed.<br />
(ii) Let d be <strong>the</strong> integer such that s (m)<br />
d<br />
≥ pn , but s (m)<br />
d−1 < pn . To<br />
prove that d + 1 is an upper bound on dl L (F G) it is sufficient to show<br />
that<br />
δ (l+1) (F G) ⊆ I(G ′ ) s(m)<br />
l for all l ≥ 0.<br />
This is clear for l = 0, and assuming that δ (l) (F G) ⊆ I(G ′ ) s(m)<br />
l−1, by<br />
Lemma 3.2.2 we obtain<br />
δ (l+1) (F G) = [δ (l) (F G), δ (l) (F G)]F G<br />
⊆ [I(G ′ ) s(m)<br />
l−1 , I(G ′ ) s(m)<br />
l−1]F G ⊆ I(G ′ ) s(m)<br />
l .<br />
Therefore, dl L (F G) ≤ d + 1. Let us choose an element aC <strong>of</strong> order<br />
2 m in G/C, set x k = x a and consider <strong>the</strong> group H = 〈x, a〉. Since<br />
(x, a) = x −1+k ∈ H ′ and k ≡ 1 (mod p), we have that H ′ has order<br />
p n . Moreover, H/CH(H ′ ) has also order 2 m . If m = 1 <strong>the</strong>n, as<br />
we have already seen before, H = H/ζ(H) is isomorphic to <strong>the</strong> dihedral<br />
group <strong>of</strong> order 2p n , so <strong>the</strong> statement is a consequence <strong>of</strong> Lemma