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Examples of nil-algebras and of infinite groups. (Exemples de nil-algèbres et de groupes infinis.) (French) Zbl 1073.16503

For any field \(K\) and for any integer \(d\geq 2\), a Golod algebra \(A\) is a non-nilpotent \(d\)-algebra (algebra with \(d\) generators) such that every \((d-1)\)-subalgebra is nilpotent. Analogous results for the Lie algebra \(AL\) which is generated by fixed generators \(X_1,\dots,X_d\) of \(A\) and for the subgroup \(G=\langle 1+X_1,\dots,1+X_d\rangle\) of the \(p\)-group \(1+A\), where 1 is the unity of \(K\), were derived by E. S. Golod [Am. Math. Soc., Transl., II. Ser. 84, 83-88 (1969); translation from Tr. Mezdunarod. Kongr. Mat., Moskva 1966, 284-289 (1968; Zbl 0206.32402)]. \(AL\) and \(G\) are called the Golod-Lie algebra and the Golod group respectively. An algebra is nil [resp., \(n\)-nilpotent] if every element is nilpotent [resp., every set of \(n\) elements generates a nilpotent subalgebra]. A. I. Sozutov [Algebra Logic 30, No. 1, 70-72 (1991); translation from Algebra Logika 30, No. 1, 102-105 (1991; Zbl 0820.20031)] constructed, for every \(k\) in a fixed finite set of integers and for any finite field, a non-nilpotent nil-\(d\)-algebra such that any \(d^k-1\) elements of minimum degree \(k\) generate a nilpotent subalgebra. Let \(M=\{a_1,a_2,\dots,a_k,\dots\}\) be a set of positive integers for which there exists a real \(0<e<1/2\) such that \(a_{i-1}\leq a_i<(d-2e)^i\) for all \(i\geq 1\) and \(a_0=1\). The main result of this paper is: For all \(d\geq 2\) and on all fields, there exists a residually finite nil-\(d\)-algebra of infinite dimension such that for all \(a_t\in M\) one has: for all families of \(a_t\) polynomials of minimum degree greater than or equal to \(t\), there exists an integer \(h\) (depending solely on \(t\) and the maximum degree of these elements) such that all products of length \(h\) of these \(a_t\) polynomials are nul. This result enables the author to construct conjugate biprimitively nilpotent groups that are not biprimitively nilpotent. The paper concludes with some theorems on decompositions.

MSC:

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
20E25 Local properties of groups
20F18 Nilpotent groups
20F50 Periodic groups; locally finite groups
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References:

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