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Transactions of the American Mathematical Society

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The action of the automorphism group of $ F\sb{2}$ upon the $ A\sb{6}$- and $ {\rm PSL}(2,\,7)$-defining subgroups of $ F\sb{2}$


Author: Daniel Stork
Journal: Trans. Amer. Math. Soc. 172 (1972), 111-117
MSC: Primary 20E05
MathSciNet review: 0310060
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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 [Enhancements On Off] (What's this?)

  • [1] Daniel F. Stork, Structure and applications of Schreier coset graphs, Comm. Pure Appl. Math. 24 (1971), 797–805. MR 0294478
  • [2] P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134-151.
  • [3] Helmut Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775

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DOI: https://doi.org/10.1090/S0002-9947-1972-0310060-1
Keywords: Automorphisms, free group, coset graph, alternating group, linear fractional transformation
Article copyright: © Copyright 1972 American Mathematical Society