Effectiveness of Walker's cancellation theorem

Pub Date : 2024-09-13 DOI:10.1002/malq.202400030
Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie
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引用次数: 0

Abstract

Walker's cancellation theorem for abelian groups tells us that if A $A$ is finitely generated and G $G$ and H $H$ are such that A G A H $A \oplus G \cong A \oplus H$ , then G H $G \cong H$ . Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between G $G$ and H $H$ , given indices for A $A$ , G $G$ , H $H$ , the isomorphism between A G $A \oplus G$ and A H $A \oplus H$ , and the rank of A $A$ , is 0 $\mathbf {0^{\prime }}$ . Moreover, we find that the complexity remains 0 $\mathbf {0^{\prime }}$ even if the generators in the copies of A $A$ are specified.

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沃克取消定理的有效性
沃克的无边际群取消定理告诉我们,如果 是有限生成的 , 且 , 那么 。德沃(Deveau)指出,该定理可以被有效化,但不是均匀地有效化。在本文中,我们对 Deveau 的初步分析进行了扩展,证明在给定 、 、 、 之间同构的指数以及 、 的秩的情况下,统一输出 、 与 之间同构的指数的复杂度为 。此外,我们还发现,即使指定了 和 的副本中的生成器,复杂度依然存在。
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