Bol Loops of Order 16

This site lists all 2052 (right) Bol Loops of order 16. I'll provide links to classifications of Bol loops of other small orders as they become available. I would appreciate an email message () from you if you have any comments regarding this work.

I have made available a quick review of the relevant definitions; for further explanation, proofs, etc., please consult a suitable list of references.

There are (to within isomorphism) exactly 2052 (right) Bol loops of order 16 (including 14 groups and 2038 non-associative loops, of which 5 are non-associative Moufang, and 14 have trivial centre). There are 390 isotopy classes of such loops. I have made available descriptions of these loops in html format, as well as Cayley tables in plain text form. This information can be surveyed through the tables listed below. I have also provided a cross-reference list comparing this list with previously published lists of groups and loops.

None of these loops are simple. This is clearly true of the loops with nontrivial centre. For each of the 14 loops with trivial centre, we have [L:K]=2 or 4 where K is the (unique) minimal normal subloop subject to the requirement that the quotient L/K be elementary abelian. This follows directly from the fact that the associated 3-net (or web, for our European colleagues) has 2-rank equal to 45 or 44 respectively (see my 1991 paper).

The vast majority of the loops in our list (1981 out of 2052) have |Z(L)|=2. In this case it is useful to define a graph on seven vertices (the seven nontrivial cosets of Z(L) in L) with two vertices joined iff elements of the corresponding cosets commute with each other. This graph, which we denote by Comm(L), is clearly an isomorphism invariant of L. We have listed the loops according to Comm(L) (whenever |Z(L)|=2), |I(L)| and |C(L)|. Exactly 252 graphs on 7 vertices arise as Comm(L) for some non-associative loop L having centre of order 2. If Comm(L) has more than 10 edges, then for clarity we display instead (in reverse video) the complementary graph (which then has at most 10 edges). In some cases these invariants (and those displayed in the linked pages available) are insufficient to distinguish isomorphism classes; see comments accompanying one of our tables.

In listing elements of the commutator (resp. associator) subloop of each of our loops, we have printed in italics any elements which are not actual commutators (resp. associators). (An example of this rare phenomenon is found in loop 16.9.8.3).

The 2038 non-associative loops, tabulated by number of involutions and size of centrum

The 2038 non-associative loops, tabulated by order of centre

• the 14 Bol loops of order 16 with trivial centre
• the 1978 non-associative Bol loops of order 16 with centre of order 2 (325 KB)
• the 46 non-associative Bol loops of order 16 with centre of order 4 (4 KB)

Other Lists

• the 5 nonassociative Moufang loops of order 16
• the 14 groups of order 16
• Cayley tables for all 2052 Bol loops of order 16:     zipped text file (230 KB) or gzipped text file (178 KB)
• the 390 isotopy classes of the 2052 loops

Naming of the Loops

Methodology Used

Case (i): Nontrivial Centre   If L is any Bol loop of order 16 with nontrivial centre, then for any subgroup Z of order 2 contained in Z(L), the quotient loop L/Z(L) is Bol of order 8. Since an exhaustive list of Bol loops of order 8 is known (5 groups and 6 non-associative loops; see Burn 1978 above), using the theory of central extensions we have generated all possibilities for L. Among these we have retained a single representative of each isomorphism class. To accomplish this, we have represented L by a graph (NOT the graph Comm(L) defined above, as this is not sufficient to distinguish all isomorphism classes; but rather a larger graph which encodes all information in the Cayley table of L). I have then used Brendan McKay's software package nauty to find a "canonical" representative for each resulting graph.

Case (ii): Trivial Centre   In this case we use a backtrack search. Classification of the resulting loops is greatly simplified since we reject here those with nontrivial centre (the vast majority of loops generated). Our basic tool is a backtrack program which reduces the search to the following cases:

• loops of exponent 2 (i.e. all elements have order 1 or 2) with trivial centre (3 loops were found in this case);
• loops of exponent 4 (i.e. all elements have order 1, 2 or 4, and having at least one element of order 4), found by completing four columns of the Cayley table generated by an element of order 4 (11 loops were found in this case); and
• loops of exponent 8 (i.e. all elements have order 1, 2, 4 or 8, and having at least one element of order 8), found by completing eight columns of the Cayley table generated by an element of order 8 (0 loops were found in this case).

I have also used the computational algebra package GAP (Graphs, Algorithms and Programming) to compute orders of left, right and full multiplication groups.

Acknowledgement

/ revised November, 2001