# Characterization of Bol Loops of Small Orders.

Thesis

Finite Bol loops of small orders are characterised in this study. There exist up to isomorphism 6 non-associative Bol loops of order 8, 2 non-associative Bol loops of order 4p and every Bol loop of order 2p or p2 is a group (where p is an odd prime). Some properties of loops satisfying identities of Bol-Moufang type are discussed. The common properties between these loops and loops satisfying Bol identity are investigated. A general construction Theorem which yields Bol loops of order 2p2 is given. It is also proved that there exist only two non-isomorphic Bol loops of order 2p2. This settles some of the open problems stated by Niederreiter, H. and Robinson, K.H. concerning existence of Bol loops of orders 18, 50 and 98. It is also proved that there are 6(p + 7) and -(p + 5), non-isomorphic Bol loops of order 3p, when 31p-1 and 31p-1 respectively and that Bol loops of order 3p are isomorphic to their loop isotopes. These results are at variance with the claims of Niederreiter, H. and Robinson, K.H. Finally, it is proved that there exist a total of 472 non-isomorphic Bol loops of order 16. This result has been verified on the computer and the relevant programmes are listed in the Appendices.