We obtain some sufficient conditions for potency and virtual potency for automorphism groups and the split extensions of some groups. In particular, considering a finitely generated group \( G \) residually \( p \)-finite for every prime \( p \), we prove that each split extension of \( G \) by a torsion-free potent group is a potent group, and if the abelianization rank of \( G \) is at most 2 then the automorphism group of \( G \) is virtually potent. As a corollary, we derive the necessary and sufficient conditions of virtual potency for certain generalized free products and HNN-extensions.

中文翻译：

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论自同构群与分裂扩张的虚势

我们得到了自同构群和某些群的分裂扩张的效价和虚效价的一些充分条件。特别地，考虑到 对于每个素数 \( p \) ，有限生成群 \ ( G \) 残差 \( p \) -有限，我们证明\( G \)的每个无扭转有效群的分裂扩展是一个有效群，如果\( G \)的阿贝尔化秩至多为 2，那么\( G \)的自同构群实际上是有效的。作为推论，我们推导出某些广义自由产品和 HNN 扩展的虚拟效力的充分必要条件。