We show that the group of bounded sequences of elements of $\mathbb{Z}[\sqrt 2]$ is an example of an abelian group with several well known, and not so well known, pathological properties. It appears to be simpler than all previously known examples for some of these properties, and at least simpler to describe for others.

中文翻译：

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病理阿贝尔群：一个友好的例子

我们证明 $\mathbb{Z}[\sqrt 2]$ 的元素有界序列群是一个阿贝尔群的例子，它具有几个众所周知的，但不太为人所知的病理特性。对于这些属性中的一些属性，它似乎比所有以前已知的示例更简单，并且对于其他属性的描述至少更简单。