On the Derived Length of Lie Solvable Group Algebras

4 CHAPTER 1
is the so-called augmentation map, and its kernel
ω(F G) = {x ∈ F G | ε(x) = 0}
the augmentation ideal of the group algebra F G. It is well-known that
ω(F G) is nilpotent if and only if G is a finite p-group and char(F ) = p.
Furthermore, if tN(G) denotes the nilpotency index of ω(F G) and
G has order pn , then 1 + n(p − 1) ≤ tN(G) ≤ pn . For example,
if G = 〈a1〉 × · · · × 〈an〉 and the order of ai is pmi , then tN(G) =
1 + n i=1 (pmi − 1).
For any normal subgroup H of G the set
I(H) = {(h − 1)x | h ∈ H, x ∈ F G}
is a two-sided ideal of F G. Clearly, I(G) coincides with ω(F G) and
I(H) = ω(F H)F G. Let T (G/H) be a transversal of the normal subgroup
H in G. Then all the elements of the form (h − 1)u, where
1 = h ∈ H and u ∈ T (G/H) form a basis of the vector space I(H).
The isomorphism F G/I(H) ∼ = F (G/H) is valid, which is called the
isomorphism theorem of group algebras.
We shall use the following notations. For x, y, x1, x2, . . . , xn ∈ G
let xy = y−1xy, (x, y) = x−1xy , and the commutator (x1, x2, . . . , xn) is
defined inductively to be
(x1, x2, . . . , xn−1), xn . We shall use freely
the commutator identities
and the Hall-Witt identity
(a, bc) = (a, c)(a, b) c = (a, c)(a, b)(a, b, c);
(ab, c) = (a, c) b (b, c) = (a, c)(a, c, b)(b, c),
(a, b −1 , c) b (b, c −1 , a) c (c, a −1 , b) a = 1,
for any a, b, c ∈ G.
Our group theoretical notation is mostly standard. By ζ(G) we
mean the center, by aut(G) the automorphism group of the group G,
by Syl p(G) the p-Sylow subgroup, by exp(G) the exponent and by
γn(G) the n-th term of the lower central series of G with γ1(G) = G.