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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On subgroups of $ M\sb{24}$. I. Stabilizers of subsets

Author: Chang Choi
Journal: Trans. Amer. Math. Soc. 167 (1972), 1-27
MSC: Primary 20B20
MathSciNet review: 0294472
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Abstract: In this paper we study the orbits of the Mathieu group $ {M_{24}}$ on sets of n points, $ 1 \leqq n \leqq 12$. For $ n \geqq 6,{M_{24}}$ is not transitive on these sets, so we may classify the sets into types corresponding to the orbits of $ {M_{24}}$ and then show how to construct a set of each type from smaller sets. We determine the stabilizer of a set of each type and describe its representation on the 24 points. From the conclusions, the class of subgroups which are maximal among the intransitives of $ {M_{24}}$ can be read off. This work forms the first part of a study which yields, in particular, a complete list of the primitive representations of $ {M_{24}}$.

References [Enhancements On Off] (What's this?)

  • [1] Chang Choi, The subgroup structure of the Mathieu group of degree 24, Ph.D. Thesis, University of Michigan, Ann Arbor, Mich., 1968.
  • [2] F. N. Cole, The transitive substitution groups of nine letters, Quart. J. Math. 26 (1893), 250-258. MR 1557254
  • [3] -, List of the transitive substitution groups of ten and of eleven letters, Quart. J. Math. 27 (1895), 39-50.
  • [4] G. Frobenius, Über die Charaktere der mehrfach transitive Gruppen, S.-B. Preuss. Akad. Wiss. 1904, 558-571.
  • [5] M. Hall, The theory of groups, Macmillan, New York, 1959. MR 21 #1996. MR 0103215 (21:1996)
  • [6] D. Livingstone and A. Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z. 90 (1965), 393-403. MR 32 #4183. MR 0186725 (32:4183)
  • [7] G. A. Miller, Collected works. Vol. I, Univ. of Illinois Press, Urbana, Ill., 1935.
  • [8] J. A. Todd, A representation of the Mathieu group $ {M_{24}}$ as a collineation group, Ann. Mat. Pura Appl. (4) 71 (1966), 199-238. MR 34 #2713. MR 0202854 (34:2713)
  • [9] 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)
  • [10] E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg 12 (1937), 256-264.

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Keywords: Mathieu groups, 5-fold transitive permutation groups, stabilizers of subsets
Article copyright: © Copyright 1972 American Mathematical Society

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