Self-invariant \(1\)-factorizations of complete graphs and finite Bol loops of exponent \(2\). (Q846759)

An example of a not soluble finite simple Bol loop of exponent 2 is presented and investigated. This example is called the Aschbacher loop \(A\). \(A\) is a simple loop of order 96 with automorphism group of size 80. This theorem implies the following result: Let \(\Omega\) be a complete graph on 96 vertices and assume that \(K\) is a 1-factorization of \(\Omega\). Put \(G_K=\langle K\rangle\). Then there exists a self-invariant factorization of \(\Omega\) such that \(G_K\) is a non-split extension \(2^{5\cdot}\text{PGL}_2(5)\). Moreover, it is shown: If \(B\) is a finite Bol loop of exponent 2 which is not isomorphic to \(A\) and which is not soluble, then \(|A|<|B|\).