| The numbering of groups in this section is different from that in later sections and in the current Groups32 program. (Some of the groups in this section are isomorphic.) |
Groups16 is an early version of the groups system. The system originated with a posting to Usenet in 1989 by Kenneth Almquist. He had generated tables for groups of orders 1-16, apparently as an exercise in combinatorics programming. In his original posting some of the tables represented isomorphic groups. Groups16 started in an effort to find out how much (or how little) one must know about groups to distinguish which tables were isomorphic.
The Almquist tables looked like this:
1 group of order 1: ___ A|A
1 group of order 2: _____ A|A B B|B A
1 group of order 3: _______ A|A B C B|B C A C|C A B
2 groups of order 4: _________ _________ A|A B C D A|A B C D B|B A D C B|B C D A C|C D A B C|C D A B D|D C B A D|D A B C
1 group of order 5: ___________ A|A B C D E B|B C D E A C|C D E A B D|D E A B C E|E A B C D
An interesting thing happened: Some readers pointed out that Almquist had posted more groups of certain orders than the literature claimed. He traced the problem to the fact that his algorithm had not detected that some of the original tables were isomorphic. A subsequent posting corrected the error. However, this raised an interesting question: How much do you need to know about a group of order 1-16 to be able to easily detect isomorphism? It is obvious that, even for groups of order 16, one does not try all possible bijective mappings to see if two groups are isomorphic. Instead, one hopes to look at properties of the groups that must be preserved under isomorphism. The original steps in the system were to represent the data in a suitable way, and to implement words that computed various information about the groups.