File:Criticalsubgrouptheorem.png

A diagrammatic aid to the proof of Thompson's critical subgroup theorem. The file was created using Graphviz and the generating code is:

digraph G{
        rankdir="BT";
        K[label="K, maximal among abelian characteristic subgroups (CS1)"];
        H[label="H = C_G(K), centralizer of K in G (CS3), characteristic in G (\
PS2)"];
        G[label="G, whole group"];
        L[label="L, inverse image of Omega_1(Z(G/K)) in G (CS6), characteristic\
 in G (PS1)"];
        C[label="C, intersection of H and L (CS7), characteristic in G (PS3), t\
he critical subgroup",peripheries=2];
        K -> C[label="precisely center: PS5"];
        C -> H;
        C -> L;
        K -> H[label="proper; see CS3"];
        K -> L;
        H -> G;
        L -> G;
}