File:Replacement2mod.png

This file was created as a diagram aid for the proof of Thompson's replacement theorem for abelian subgroups. The file was created using Graphviz with the following code:

digraph G{       rankdir="BT"; A[label="A, abelian of maximum order"]; B[label="B, abelian, normalized by but does not normalize A"]; C[label="C = AB"]; N[label="N = N_B(A)"]; AcapB[label="Intersection of A and B"]; BmodNcapZCmodNinverseimage[label="inverse image mod N of B/N intersect Z(C/N)"]; M[label="M = [x,A], x nontrivial in inverse image mod N of B/N intersect Z(C/N)"]; Astar[label="A* = MC_A(M), claimed replacement",peripheries=2]; AstarcapB[label="Intersection of A* and B"]; CAM[label="C_A(M), centralizer in A of M"]; P[label="P, whole group"]; T[label="trivial subgroup"]; T -> AcapB; AcapB -> N[label="proper"]; AcapB -> A;       N -> BmodNcapZCmodNinverseimage[label="proper"]; BmodNcapZCmodNinverseimage -> B;       A -> C;        B -> C;        C -> P;        T -> M -> N;        AcapB -> CAM -> A;        M -> Astar; CAM -> Astar; AcapB -> AstarcapB[label="proper"]; AstarcapB -> B;       AstarcapB -> Astar; Astar -> C; }