permutation group

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permutation group

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permutation group (plural permutation groups)

1. (algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.
   • 1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80:

     In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.

   • 1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47:

     The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).

   • 2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86:

     Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group. […] The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.

• (group whose elements are permutations of a set): transformation group

• (group whose elements are permutations of a set): alternating group, symmetric group

• permutation representation