A finite group of order $n$ can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order $n$ can be stored using $O(n^2)$ words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order $n$ that uses $o(n^2)$ space but can still answer a multiplication query in constant time. We design a constant query-time data structure that can store any finite group using $O(n)$ words where $n$ is the order of the group. Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of $\Omega(n)$ on the number of words to store a group of order $n$. Since our data structure achieves this lower bound and answers queries in constant time, it is optimal in both space usage and query-time. A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonableian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups (CFSG).

翻译：有限群$\mathrm{order}(n)$可通过Cayley表表示。在字RAM模型中，$\mathrm{order}(n)$的群Cayley表可用$O(n^2)$个字存储，并能在常数时间内回答乘法查询。一个有趣的问题是：能否设计一种数据结构，在仅使用$o(n^2)$空间的情况下，仍能对$\mathrm{order}(n)$的群实现常数时间乘法查询？本文设计了一种常数查询时间的数据结构，可用$O(n)$个字存储任意有限群，其中$n$为群阶。Farzan与Munro（ISSAC 2006）给出了存储$\mathrm{order}(n)$群所需字数的信息论下界$\Omega(n)$。由于我们的数据结构达到该下界且能在常数时间内回答查询，因此在空间使用与查询时间上均为最优。该过程中的关键步骤是为非阿贝尔单群设计线性空间与常数查询时间的数据结构。针对非阿贝尔单群的数据结构基于一个引理设计，该引理通过有限单群分类定理（CFSG）证明。