Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Finite groups and semigroups

Any finite group or semigroup can be thought of as defined by having a multiplication table which satisfies the constraints given on page 887. The total number of semigroups increases faster than exponentially with size in a seemingly quite uniform way. But the number of groups varies in a complicated way with size, as in the picture below. (The peaks are known to grow roughly like n^(2/27 Log[2, n]^2)—intermediate between polynomial and exponential.) As mentioned on page 938, through major mathematical effort, a complete classification of all finite so-called simple groups that in effect have no factors is known. Most such groups come in families that are easy to characterize; a handful of so-called sporadic ones are much more difficult to find. But this classification does not immediately provide a practical way to enumerate all possible groups. (See also pages 938 and 1032.)

From Stephen Wolfram: A New Kind of Science [citation]