Linear Space Data Structures for Finite Groups with Constant Query-Time

Abstract

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. Das et al. (J Comput Syst Sci 114:137–146, 2020) showed that for any finite group G of order n and for any \(\delta \in [1/\log {n}, 1]\), a data structure can be constructed for G that uses \(O(n^{1+\delta }/\delta )\) space and answers a multiplication query in time \(O(1/\delta )\). 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. 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. Since our data structure achieves the information theoretic 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 nonabelian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups.