We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

中文翻译：

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关于有限群的一个语言子群的等级

我们证明，如果w是一个多线性交换子词，而G是一个有限群，其中由w值生成的每个变幂次子群的秩最多为r，那么语言子群 $w(G)$ 的秩是有界的只有r和w。在G是可溶的情况下，我们得到更好的结果：如果G是一个有限可溶群，其中由w值生成的每个幂等子群的秩最多为r，那么 $w(G)$ 的秩为大多数 $r+1$ 。