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Permutation Groups and Group Actions

In: Algebra

Author

Listed:
  • Ernest Shult

    (Kansas State University, Department of Mathematics)

  • David Surowski

Abstract

A useful paradigm for discussing a group is to regard it as acting as a group of permutations of some set. The power of this point of view derives from the flexibility one has in choosing the set being acted on. Odd and even finitary permutations, the cycle notation, orbits, the basic relation between transitive actions and actions on cosets of a subgroup are first reviewed. For finite groups, the paradigm produces Sylow’s theorem, the Burnside transfer and fusion theorems, and the calculations of the order of any group of automorphisms of a finite object. Of more special interest are primitive and multiply transitive groups.

Suggested Citation

  • Ernest Shult & David Surowski, 2015. "Permutation Groups and Group Actions," Springer Books, in: Algebra, edition 127, chapter 0, pages 105-136, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-19734-0_4
    DOI: 10.1007/978-3-319-19734-0_4
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