A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G is an element of X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. It is still an open problem whether every word is concise in the class of residually finite groups. A word w is rational if the number of solutions to the equation w(x(1), . . ., x(k)) = g is the same as the number of solutions to w(x(1), . . ., x(k)) = g(e) for every finite group G and for every e relatively prime to G. We observe that any rational word is concise in the class of residually finite groups. Further we give a sufficient condition for rationality of a word. As a corollary we deduce that the word w = [. . .[x(1)(n1), x(2)](n2), . . ., x(k)](nk) is concise in the class of residually finite groups. (C) 2015 Elsevier Inc. All rights reserved.
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机译：如果组G中w值的集合是有限的，则总是将词组w（G）定义为有限的，因此将组词w称为简洁。更一般而言，如果每当组G的w值是有限的时，单词w就被简明表示为组G，则总是遵循w（G）是有限的。 P. Hall问每个词是否简洁。由于伊万诺夫（Ivanov），这个问题的答案是负面的。每个单词在剩余有限组的类别中是否简洁，仍然是一个悬而未决的问题。如果方程w（x（1），...，x（k））= g的解数与w（x（1），...，...的解数相同，则单词w是有理的。 ，对于每个有限群G和相对于 G 的每个e，x（k））= g（e）。我们观察到任何有理词在剩余有限群类中都是简洁的。此外，我们为单词的合理性提供了充分的条件。作为推论，我们推导出单词w = [。 。 。[x（1）（n1），x（2）]（n2），。 。 。，x（k）]（nk）在残差有限群的类中很简洁。 （C）2015 Elsevier Inc.保留所有权利。
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