In this paper, we consider the influence that the maximal size $m$ of an abelian subgroup of a group exerts on the size of the group. We will first prove that $|G|$ divides $g(m)$, the product of all prime powers at most $m$. We then show that if a prime $p > m/2$ divides $|G|$ then either $G$ is almost simple or of very restricted type and we determine the complete list of finite simple groups with exactly one such "large" prime divisor. We are then able to deduce that $|G|=g(m)$ holds only when $G$ is a small symmetric group and to derive an explicit upper bound for $|G|$ as a function of $m$. We conclude our paper by determining the order of magnitude of this upper bound.

中文翻译：

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关于最大阶阿贝尔群的一个注记

在本文中，我们考虑了一个组的一个阿贝尔群的最大大小$ m $对该组大小的影响。我们将首先证明$ | G | $除以$ g（m）$，即所有主要幂的乘积最多$ m $。然后，我们证明如果素数$ p> m / 2 $除以$ | G | $，则$ G $几乎是简单的，或者是非常受限制的类型，并且我们确定了带有一个这样的“大”的有限简单组的完整列表。本数除数。然后，我们可以推断出仅当$ G $是一个小的对称组时，$ | G | = g（m）$成立，并得出$ | G | $作为$ m $的函数的明确上限。我们通过确定该上限的数量级来结束本文。