### Quasiprimitivity and quasigroups

J.D. Phillips, J.D.H. Smith
1999 Bulletin of the Australian Mathematical Society
It is well known that Q is a simple quasigroup if and only if Mlt Q acts primitively on Q. Here we show that Q is a simple quasigroup if and only if MltQ acts quasiprimitively on Q, and that Q is a simple right quasigroup if and only if RMlt Q acts quasiprimitively on Q. A quasigroup is set with a single binary operation, denoted by juxtaposition, such that in xy = z, knowledge of any two of x,y and z specifies the third uniquely. A right quasigroup is a set with a single binary operation whose
more » ... right translations biject. The multiplication group, Mlt Q, of a quasigroup Q is the subgroup of the group of all bijections on Q generated by right and left translations, that is MltQ : of a right quasigroup Q is the subgroup of the group of all bijections on Q generated by right translations, that is RMltQ :-(R(x) : x 6 Q). A quasigroup Q is called type 1 if RMltQ = MltQ. For example, commutative quasigroups are type 1; so too are finite simple Moufang loops [4]-A permutation group G on a set Q acts primitively on Q if the only G-invariant partitions of Q are the two trivial partitions {Q} and {{x} : x 6 Q}. Of course, if G acts primitively on Q then each nontrivial normal subgroup of G is transitive on Q. In [5], Praeger used this fact to generalise the notion of primitivity: a permutation group G on a set Q acts quasiprimitively on Q if each non-trivial normal subgroup of G is transitive on Q. This definition is useful because, as Praeger proved [5], there is an O'Nan-Scott type theorem classifying all quasiprimitive permutation groups as one of eight types. Given a quasigroup Q, there exist two binary operations /, \ on Q such that (xy)/y = x, (x/y)y -x, x\(xy) = y, and x(x\y) = y. Conversely, an algebra with three binary operations satisfying these four identities is a quasigroup -as defined at the beginning of this paper -under any one of these operations [2] . Similarly, right quasigroups are axiomatised by the first two of the four quasigroup identities above. A