The Cayley table representation of a group uses $\mathcal{O}(n^2)$ words for a group of order $n$ and answers multiplication queries in time $\mathcal{O}(1)$. It is interesting to ask if there is a $o(n^2)$ space representation of groups that still has $\mathcal{O}(1)$ query-time. We show that for any $\delta$, $\frac{1}{\log n} \le \delta \le 1$, there is an $\mathcal{O}(\frac{n^{1 +\delta}}{\delta})$ space representation for groups of order $n$ with $\mathcal{O}(\frac{1}{\delta})$ query-time. We also show that for Z-groups, simple groups and several group classes defined in terms of semidirect product, there are linear space representations with at most logarithmic query-time. Farzan and Munro (ISSAC'06) defined a model for group representation and gave a succinct data structure for abelian groups with constant query-time. They asked if their result can be extended to categorically larger group classes. We construct data structures in their model for Hamiltonian groups and some other classes of groups with constant query-time.

中文翻译：

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有限群的空间有效表示

一个组的凯莱表表示使用 $\mathcal{O}(n^2)$ 词来表示一组 $n$ 并在 $\mathcal{O}(1)$ 时间内回答乘法查询。询问是否有 $o(n^2)$ 空间表示仍然具有 $\mathcal{O}(1)$ 查询时间的组。我们证明对于任何 $\delta$, $\frac{1}{\log n} \le \delta \le 1$, 有一个 $\mathcal{O}(\frac{n^{1 +\delta }}{\delta})$ 空间表示为 $n$ 阶组，$\mathcal{O}(\frac{1}{\delta})$ 查询时间。我们还表明，对于根据半直接积定义的 Z 群、简单群和几个群类，存在最多对数查询时间的线性空间表示。Farzan 和 Munro (ISSAC'06) 定义了一个群表示模型，并为具有恒定查询时间的阿贝尔群提供了一个简洁的数据结构。他们询问他们的结果是否可以扩展到绝对更大的组类。我们在他们的模型中为哈密顿群和其他一些具有恒定查询时间的群类构建数据结构。