Succinct Representations of Finite Groups

The Cayley table representation of a group uses (formula presented) words for a group of order n and answers multiplication queries in time O(1) in word RAM model. It is interesting to ask if there is a (formula presented) space representation of groups that still has O(1) query-time. We show that for any (formula presented), (formula presented), there is an (formula presented) space representation for groups of order n with (formula presented) query-time. We also show that for Dedekind groups, simple groups and several group classes defined in terms of semidirect product, there are linear space representation to answer multiplication queries in logarithmic 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 show we can construct data structures in their model to represent Hamiltonian groups and extensions of abelian groups by cyclic groups to answer multiplication queries in constant time.