By G. Viennot
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Extra resources for Algebres de Lie Libres et Monoides Libres
U, F(X)] Thus D = F(X) and = 1. Now any Sylow 2-subgroup of U centralizes F(O(X)) ~ F(X). 2 it follows that any Sylow 2-subgroup of U centralizes O(X). 1 it follows that any Sylow 2-sub- group of U lies in F*(X) and so U;; F*(X). 3(a), U = [t, U] <]<] E(X). Thus U ~ E(X). 3(b) is not applicable here since if a component K of U is not a component of E (X), then [K, t] = 1. But U = [t, U] is just the product of components not centralized by t. Now assume that U = [t, U] is a p-group where p is an odd prime and that X is a minimal counter example.
We show first that Then [CF(X)(t), U] ~ U n F(X), a nilpotent normal subgroup of U and so [U, Cx(t) n F(X)) ~ Z(U). usual, [U, Cx(t) n F(X)] = 1 and so Cx(t) r, F(X) ~ D. (NX(D)nF(X));D. As Thus t inverts Note that a Sylow 2-subgroup of F(X) lies in CX(t) n F(X) ~ D. Now U = [t, U] centralizes NX(D) n F(X);D. Arguing for each Sylow p-subgroup of the nilpotent group NX(D) n F(X) separately, it follows that U centralizes NX(D) n F(X). [U, F(X)] Thus D = F(X) and = 1. Now any Sylow 2-subgroup of U centralizes F(O(X)) ~ F(X).
They are also of interest because they seem not to have been noticed in even the solvable case. 5. If G is a group such that F*(G) is a p-group. If U is a p-subgroup of G, then F*(CG(U)) and F*(NG(U)) are p- gr oups als o. Let N == NG(U), C == CG(U). Proof. F(N)p' and E(N) ~ C because U ~ F(N). 6 E(N) == E(C). It follows that OP(F*(N)) == oP(F*(C)). Let x oP(F*(N)) be a pt-element. E Then [cG(u)rF*(G), x];; OP(F*(N)) n F*(G) ~ Z(F*(N)). Hence [CG(U)nF*(G), x, x) == 1. 2, [cG(u)nF*(G), x) = 1. 2 to
Algebres de Lie Libres et Monoides Libres by G. Viennot