The action of the automorphism group of $F_{2}$ upon the $A_{6}$- and $\textrm {PSL}(2, 7)$-defining subgroups of $F_{2}$

Abstract:

In this paper is described a graphical technique for determining the action of the automorphism group ${\Phi _2}$, of the free group ${F_2}$ of rank 2 upon those normal subgroups of ${F_2}$ with quotient groups isomorphic to $G$, where $G$ is a group represented faithfully as a permutation group. The procedure is applied with $G = {\text {PSL}}(2,7)$ and ${A_6}$ (the case $G = {A_5}$ having been treated in an earlier paper) with the following results: Theorem 1. ${\Phi _2}$, acts upon the 57 subgroups of ${F_2}$ with quotient isomorphic to ${\text {PSL}}(2,7)$ with orbits of lengths 7, 16, 16, and 18. The action of ${\Phi _2}$ is that of ${A_{16}}$ in one orbit of length 16, and of symmetric groups of appropriate degree in the other three orbits. Theorem 2. ${\Phi _2}$, acts upon the 53 subgroups of ${F_2}$ with quotients isomorphic to ${A_6}$ with orbits of lengths 10, 12, 15, and 16. The action is that of full symmetric groups of appropriate degree in all orbits.