The Normality Of The Connected Cubic Coset Graphs On Finite Groups

In studing the groups and graphs, the symmetry property of the graphs has been being a very hot issue, which is mainly described by some transitive properties of their automorphism groups. The classical representations of these type graphs are Cayleygraphs and Sabidussi Coset graphs.In fact, researching the symmetry property of the Coset graph is more significant than the Cayleygraph, because every vertex-transitive graph is always some Coset graph of its full automorphism group. Similar to the normality of the Cayleygraph, we can also define the normality of the Coset graph, that is, if the relevant group is normal to its full automorphism of the Coset graph. Further, we can also define if a Coset graph is a graph representation, we note it GR for short, that is, the relevant group is exactly equal to the full automorphism group of the Coset graph.Caiheng Li has proved that[1]: the connected cubic arc-transitive Cayleygraphs on the finite nonabelian simple groups are all normal except A₅, PSL(2,11), M₁₁, A₁₁, M₂₃, A(23) and A₄₇. Shangjin Xu has proved that [2]: the connected cubic arc-transitive Cayleygraphs on the simple groups A₅, PSL(2,11), M₁₁, A₁₁, M₂₃, A₂₃ are also normal except A₄₇. As for the left alternating group A₄₇, Shangjin Xu has structurized that its nonnormal connected cubic arc-transitive Cayleygraph, and has proved that under the meaning of isomorphism this kind of graph has just two, and both of them are 5-arc-transitive. Thus, the connected cubic arc-transitive Cayleygraphs on the finitenonabelian simple groups are all normal except two nonnormal arc-transitive graphsof order (47!)/2. This consequence not only solve the normality problem but also equiv-alently give a complete classification of the nonnormal connected cubic arc-transitive Cayleygraphs on the finite nonabelian simple groups. However, as for researching about the normality of the Coset graphs are rather few. At present, only my supervisor Shangjin Xu has studied the normality of the Coset graphs on the finite nonabelian simple groups, and he has given quit good conclusions. He has proved that most, if not all, the connected cubic G-arc-transitive Coset graphs on the finite nonabelian simple groups are normal, further, has given bounds of the full automorphism on this kind of graphs. The particular conclusions are as follows:...