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A radical for free groups. (English) Zbl 0585.20028
If G is any group, \(G_ i\) denotes the ith term of the lower central series of G. The reviewer showed that there is a useful epimorphism \(\otimes (G/G_ 2)^ n\to G_ n/G_{n+1}.\) In this paper the kernel of the epimorphism is computed in the case where G is a free group of finite rank.
Let T and L be respectively the free associative algebra and the free Lie ring on a set of d elements. Then \(L\simeq \oplus_{n\geq 1}F_ n/F_{n+1}\) where F is a free group of rank d, while T is essentially the tensor algebra on d elements. The problem is equivalent to computing the kernel of the natural epimorphism \(T_ n\to L_ n\) of homogeneous components of degree n. This turns out to be the right ideal generated by a natural set of elements.
Reviewer: D.J.S.Robinson
MSC:
20E05 Free nonabelian groups
20F14 Derived series, central series, and generalizations for groups
20E07 Subgroup theorems; subgroup growth
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