Space Efficient Representations of Finite Groups

Abstract: 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.