Permutation Group

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A permutation group is a finite group 👁 G
whose elements are permutations of a given set and whose group operation is composition of permutations in 👁 G
. Permutation groups have orders dividing 👁 n!
.

Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.

Permutation groups are represented in the Wolfram Language as a set of permutation cycles with . A set of permutations may be tested to see if it forms a permutation group using [l] in the Wolfram Language package .

Conjugacy classes of elements which are interchanged in a permutation group are called permutation cycles.

Examples of permutation groups include the symmetric group 👁 S_n
(of order 👁 n!
), the alternating group 👁 A_n
(of order 👁 n!/2
for 👁 n>=2
), the cyclic group 👁 C_n
(of order 👁 n
), and the dihedral group 👁 D_n
(of order 👁 2n
).

See also

Alternating Group, Cayley's Group Theorem, Cycle Index, Cyclic Group, Dihedral Group, Group, Netto's Conjecture, Permutation, Permutation Cycle, Permutation Graph, Permutation Involution, Symmetric Group, Transposition

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References

Cameron, P. Permutation Groups. New York: Cambridge University Press, 1999.Furst, M.; Hopcroft, J.; and Luks, E. "Polynomial Time Algorithms for Permutation Groups." In Proc. Symp. Foundations Computer Sci. IEEE, pp. 36-41, 1980.Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984.Skiena, S. "Permutation Groups." §1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 17-26, 1990.Wielandt, H. Finite Permutation Groups. New York: Academic Press, 1964.

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Permutation Group

Cite this as:

Weisstein, Eric W. "Permutation Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PermutationGroup.html

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