Abstract

A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. In particular, Olshanskii gave an example of such a word, which we denote by \(w_o\) . The problem whether every word is concise in the class of residually finite groups remains wide open. In this note, we observe that \(w_o\) is concise in residually finite groups. Moreover, we show that \(w_o\) is strongly concise in profinite groups, that is, \(w_o(G)\) is finite whenever G is a profinite group in which \(w_o\) takes less than \(2^{\aleph _0}\) values.

中文翻译：

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论Olshanskii例子中单词的简洁性

摘要

如果当w只取组G中的有限多个值时，动词子组w ( G ) 是有限的，则组词w被称为简洁的。众所周知，有些词不简洁。Olshanskii 特别给出了这样一个单词的例子，我们用\(w_o\)表示。在残差有限群类中每个单词是否简洁的问题仍然是一个悬而未决的问题。在这篇文章中，我们观察到\(w_o\)在剩余有限群中是简洁的。此外，我们证明\(w_o\)在有限群中是强简洁的，也就是说，只要G是有限群，其中\(w_o\)小于\(2^)，\(w_o(G)\)就是有限的{\aleph _0}\)值。