# Bol Loops of Order 15

I've been working on a classification of Bol loops of small order.
Here is a list of the 3 Bol loops of order 15. This includes the cyclic group of order 15 and the
2 non-associative Bol loops of order 15 (neither of which is Moufang). The two non-associative
examples are isotopic to one another.
The completeness of this list was first shown by
Niederreiter and Robinson (1981).

I have made available

- a quick review of the relevant definitions;
- a suitable list of references
for further explanation, proofs, etc.;
- a cross-reference list comparing this list with previously
published lists of groups and loops;
- descriptions of these loops in
`html` format, found through the
links below; and
- the Cayley tables (2 KB) in plain text form.

In listing elements of the commutator (resp. associator) subloop of each of these loops, we have printed *in italics*
any elements which are not actual commutators (resp. associators). (I haven't checked, however, whether in fact this
phenomenon occurs among any the loops of order 15 in our list.)

**Isotopy Class 0: The Cyclic Group**
15.2.15.0

**Isotopy Class 1: The 2 Non-Moufang (non-associative) Bol Loops**
15.10.1.0,
15.10.1.1

### Naming of the Loops

For each of the loops of order 15, I have used a name 15.*i*.*c*.*k* where *i* is the number of 3-elements,
*c*=|*C*(*L*)| and the index *k*=0,1,2,... indicates merely the order in which each isomorphism
class of loop was first encountered by my computer.

/
revised March, 2004