Let \({\cal K}\) be a root class of groups closed under taking quotient groups, G be a free product of groups A and B with amalgamated subgroups H and K. Let also H be normal in A, K be normal in B, and Aut_G(H) denote the set of automorphisms of H induced by all inner automorphisms of G. We prove a criterion for G to be residually a \({\cal K}\)-group provided Aut_G(H) is an abelian group or it satisfies some other conditions. We apply this result in the cases when A and B are bounded nilpotent groups or A/H, B/K ∈ \({\cal K}\).

中文翻译：

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关于具有正常合并子组的组的某些自由产品的根类残差

设\（{\ cal K} \）为在取商组下封闭的组的根类，G为组A和B与H和K合并的子集的自由乘积。设H在A中是正常的，K是正常的在乙，和自_ģ（ħ）表示该组构的ħ由内的所有构诱导ģ。我们证明了G的准则是残存的\（{\ cal K} \）- group，前提是Aut _G（H）是一个阿贝尔群或它满足其他一些条件。我们将这个结果应用于A和B是有界幂等组或A / H，B / K∈ \（{\ cal K} \）的情况。