MinimumPermutationRepresentationDegree - Maple Help

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GroupTheory

 MinimumPermutationRepresentationDegree
 compute the minimum degree of a permutation representation of a group

 Calling Sequence MinimumPermutationRepresentationDegree( G ) MinPermRepDegree( G )

Parameters

 G - a group

Description

 • Cayley's Theorem asserts that each finite group is isomorphic to a group of permutations of a finite set.  In other words, each finite group $G$ can be embedded in a symmetric group ${S}_{n}$, for some positive integer $n$.
 • The MinimumPermutationRepresentationDegree( G ) command returns the minimum degree of a faithful permutation representation for a (finite) group G.  That is the least positive integer n such that G embeds in the symmetric group of degree n.
 • You can use the alias MinPermRepDegree instead of the longer command name MinimumPermutationRepresentationDegree.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{MinPermRepDegree}\left(\mathrm{CyclicGroup}\left(12\right)\right)$
 ${7}$ (1)
 > $\mathrm{MinPermRepDegree}\left(\mathrm{GL}\left(2,5\right)\right)$
 ${24}$ (2)
 > $\mathrm{MinPermRepDegree}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${8}$ (3)
 > $\mathrm{MinPermRepDegree}\left(\mathrm{PSL}\left(5,q\right)\right)$
 ${{q}}^{{4}}{+}{{q}}^{{3}}{+}{{q}}^{{2}}{+}{q}{+}{1}$ (4)

Compatibility

 • The GroupTheory[MinimumPermutationRepresentationDegree] command was introduced in Maple 2016.