The action of the automorphism group of upon the - and -defining subgroups of
Abstract: In this paper is described a graphical technique for determining the action of the automorphism group , of the free group of rank 2 upon those normal subgroups of with quotient groups isomorphic to , where is a group represented faithfully as a permutation group. The procedure is applied with and (the case having been treated in an earlier paper) with the following results:
Theorem 1. , acts upon the 57 subgroups of with quotient isomorphic to with orbits of lengths 7, 16, 16, and 18. The action of is that of in one orbit of length 16, and of symmetric groups of appropriate degree in the other three orbits.
Theorem 2. , acts upon the 53 subgroups of with quotients isomorphic to with orbits of lengths 10, 12, 15, and 16. The action is that of full symmetric groups of appropriate degree in all orbits.
-  Daniel F. Stork, Structure and applications of Schreier coset graphs, Comm. Pure Appl. Math. 24 (1971), 797–805. MR 0294478, https://doi.org/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, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775
- D. Stork, The structure and applications of Schreier coset graphs, Comm. Pure Appl. Math. 24 (1971), 707-805. MR 0294478 (45:3548)
- P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134-151.
- H. Wielandt, Finite permutation groups, Lectures, University of Tübingen, 1954/55; English transl., Academic Press, New York, 1964. MR 32 #1252. MR 0183775 (32:1252)
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Keywords: Automorphisms, free group, coset graph, alternating group, linear fractional transformation
Article copyright: © Copyright 1972 American Mathematical Society