We can distinguish groups of order up to 15 by the orders of the elements. The only groups in Almquist’s list with the same set of element orders are groups 20 and 21 (of order 12), and we have found an explicit isomorphism between them.
Groups of order 16 provide the first counterexample to the hypothetical theorem. Groups 43 and 45 have the same numbers of elements of each order, but group 43 is abelian, while group 45 is not.
Further isomorphisms occur in Almquist’s tables among groups of order 16. There are also groups of order 16 that are not isomorphic but that agree in element orders and in commutativity (abelian or not). Thus we still do not have enough properties to classify groups of order 16 up to isomorphism.
Here are some additional, easily computed, properties that may be useful - I wrote them as experiments to find useful invariants. They do not appear in either Groups16 or Groups32, since they did not seem to be of lasting interest. They were, in fact, used to distinguish the original set of group tables.
Number of Commuting Pairs
Eventually (in Groups32) we compute the center of a group and can easily compute the size of the center. At this point we just run through all pairs of elements and count those pairs that commute:
: #Commuting ( grp -- n ) >GROUP 0 GLIMITS DO GLIMITS DO J I G* I J G* = IF 1+ THEN LOOP LOOP ;
Number of Squares
We can define #SQUARES to count how many elements in a group are squares:
CREATE SQ 20 ALLOT : #SQUARES ( grp# -- n ) >GROUP SQ 20 ERASE GLIMITS DO I I G* ID - SQ + 1 SWAP C! LOOP 0 20 0 DO SQ I + C@ + LOOP ;
If one actually wishes to see which elements are squares, the following word will display the squares:
: SQUARES ( grp# -- ) >GROUP SQ 20 ERASE GLIMITS DO I I G* ID - SQ + 1 SWAP C! LOOP 20 0 DO SQ I + C@ IF I ID + .ELE THEN LOOP ;
It turns out that, at this point, we have implemented enough properties to classify groups up to order 16. Coupled with the mechanism for exhibiting isomorphisms, we can detect which of Almquist's tables are for isomorphic groups. (There are actually only 42 groups of orders 1-16 up to isomorphism.)
The following tables were found to be isomorphic: (20,21), (33,40), (36,38), (32,39,42), (34,41). In the process, I produced a method that will quickly identify any group of order 1-16 up to isomorphism.