55 research outputs found
A note on the local nilpotence of 4-Engel groups
MSC 20F45 Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth-Bendix algorithm. In this paper we give a short hand-proof of the nilpotency of T
Ultraproducts and metastability
Given a convergence theorem in analysis, under very general conditions a
model-theoretic compactness argument implies that there is a uniform bound on
the rate of metastability. We illustrate with three examples from ergodic
theory
Combinatorial Polynomial Identity Theory
This dissertation consists of two parts. Part I examines certain Burnside-type conditions on the multiplicative semigroup of an (associative unital) algebra .
A semigroup is called -collapsing if, for every , there exist functions on the set such that \begin{center} . \end{center} If and can be chosen independently of the choice of , then satisfies a semigroup identity. A semigroup is called -rewritable if and can be taken to be permutations. Semple and Shalev extended Zelmanov\u27s Fields Medal writing solution of the Restricted Burnside Problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent.
The primary result of Part I is that the following conditions are equivalent for every algebra over an infinite field: the multiplicative semigroup of is collapsing, satisfies a multiplicative semigroup identity, and satisfies an Engel identity: . Furthermore, in this case, is locally (upper) Lie nilpotent. It is also shown that, if the multiplicative semigroup of is rewritable, then must be commutative.
In Part II of this dissertation, we study algebraic analogues to well-known problems of Philip Hall on the verbal and marginal subgroups of a group. We begin by proving two algebraic analogues of the Schur-Baer-Hall Theorem: if is a group such that G/\centre_n(G) is finite, where \centre_n(G) is the higher centre of , then the term, \g_{n+1}(G), of the lower central series of is also finite; conversely, if \g_{n+1}(G) is finite, then so is G/\centre_{2n}(G). Next, we prove results of a more general type.
Given an algebra and a polynomial , we define the verbal subspace, , of to be spanned by the set of -values in , the verbal subalgebra, \A_A(f), and the verbal ideal, \I_A(f), of to be generated by the set of -values in . We also define the marginal subspace of to be the set of all elements such that \begin{center} f(b_1,\ldots,b_{i-1},b_i+\a z,b_{i+1},\ldots,b_n)=f(b_1,\ldots,b_{i-1},b_i,b_{i+1},\ldots,b_n), \end{center} for all , , and \a \in K. Furthermore, we define the marginal subalgebra, \widehat{\A}_A(f), and the marginal ideal, \widehat{\I}_A(f), to be the largest subalgebra, respectively, largest ideal, of contained in . We consider the following problems: \begin{enumerate} \item If is of finite codimension in , is finite-dimensional? \item If is finite-dimensional, is of finite codimension in ? \item If is finite-dimensional, is \A_A(f) or \I_A(f) finite-dimensional? \item If is finite-dimensional, is A/\widehat{\A}_A(f) or A/\widehat{\I}_A(f) finite-dimensional? \end{enumerate
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