关于满足Engel型恒等式的群簇
On varieties of groups satisfying an Engel type identity
摘要：
设m，n是正整数，v是多线性换位子字，w = v^m.用v（G）和w（G）表示群G中分别对应于v和w的动词子群。本文证明了群G的w-值为n-Engel且w（G）局部幂零的类是一个簇（定理A）。进一步，我们证明了在m是素数幂的情形下，所有群G的w-值是n-Engel且v（G）有升正规列且其幂元是局部可解的或局部有限的类是一个簇（定理B）.我们研究的问题是否后者的结果仍然有效，允许m是一个任意的正整数。在这个方向上，我们证明了：如果m，n是正整数，u是多线性换位子字，v是896个u-字的乘积，则群G中v^m-值是n-Engel且动词子群u（G）有一个升正规列且其列元是局部可解的或局部有限的类是一个簇（定理C）.
英文摘要：
Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result remains valid with m allowed to be an arbitrary positive integer. In this direction, we show that if m, n are positive integers, u a multilinear commutator word and v the product of 896 u-words, then the class of all groups G in which the v^m-values are n-Engel and the verbal subgroup u(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem C).
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