The action of the automorphism group of $F_{2}$ upon the $A_{6}$- and $\textrm {PSL}(2, 7)$-defining subgroups of $F_{2}$
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- by Daniel Stork PDF
- Trans. Amer. Math. Soc. 172 (1972), 111-117 Request permission
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.References
- Daniel F. Stork, Structure and applications of Schreier coset graphs, Comm. Pure Appl. Math. 24 (1971), 797โ805. MR 294478, DOI 10.1002/cpa.3160240606 P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134-151.
- Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 172 (1972), 111-117
- MSC: Primary 20E05
- DOI: https://doi.org/10.1090/S0002-9947-1972-0310060-1
- MathSciNet review: 0310060