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 δ, $1/\log n\le \delta \le 1$, there is an $\mathcal{O}(n^{1+\delta }/\delta )$ space representation for groups of order n with $\mathcal{O}(1/\delta )$ query-time. We also show that for Dedekind groups, simple groups and several group classes defined in terms of semidirect product, there are $\mathcal{O}(n)$ space representations with logarithmic query-time where n is the size of the group. 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.

组的Cayley表表示使用 $\mathcal{Ø}（{ñ}^2）$一组n阶的单词并及时回答乘法查询$\mathcal{Ø}（\mathrm{1个}）$。有趣的是要问是否有$Ø（{ñ}^2）$ 仍然具有的组的空间表示 $\mathcal{Ø}（\mathrm{1个}）$查询时间。我们证明对于任何δ，$\mathrm{1个}/\mathrm{日志}ñ\le \delta \le \mathrm{1个}$， 有一个 $\mathcal{Ø}（{ñ}^{\mathrm{1个}+\delta }/\delta ）$为顺序的组空间表示Ñ与$\mathcal{Ø}（\mathrm{1个}/\delta ）$查询时间。我们还表明，对于Dedekind组，简单组和根据半直接乘积定义的几个组类，存在$\mathcal{Ø}（ñ）$具有对数查询时间的空间表示形式，其中n是组的大小。Farzan和Munro（ISSAC'06）定义了一种用于组表示的模型，并为具有恒定查询时间的阿贝尔组提供了简洁的数据结构。他们询问是否可以将其结果扩展到更大的小组类别。我们在哈密顿量组和其他具有恒定查询时间的组类别的模型中构造数据结构。