PermutationGroup

The group generated by a list of permutations (Cycles), closed by breadth-first search over products of the generators. A carrier, like Cycles — GroupOrder, GroupElements and GroupGenerators read it directly rather than it computing anything itself.

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PermutationGroup({perm, ...})the group $\langle \mathrm{perm}, \dots \rangle$ generated by the given permutations

PermutationGroup({perm, ...}, n)the same group, with its degree (largest point moved) at least n

Typed signatures

Type	Package	Applies
(list<expression<Cycles>>, integer?) -> expression<PermutationGroup>	groupalgebra	always

Domain: Permutations

Details

• An empty generator list is the trivial group, order 1
• A single generator gives the cyclic group of its order — the least common multiple of its cycle lengths
• Dihedral groups arise as PermutationGroup of an $n$-cycle (rotation) and a 2-cycle-full involution (a flip)

Examples

Basic #

lcm(3, 2) — one 3-cycle and one 2-cycle in the single generator

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In$\mathrm{GroupOrder}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 9, 6\bigr\rbrack, \bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$6$

Scope #

The trivial group has just the identity element

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In$\mathrm{GroupElements}(\mathrm{PermutationGroup}(\bigl\lbrack\bigr\rbrack))$Out$\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigr\rbrack)\bigr\rbrack$

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In$\mathrm{Same}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack\bigr\rbrack)\bigr\rbrack), \mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack4, 5\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$\mathrm{False}$

Applications #

a 4-cycle and a transposition generate all of $S_4$

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In$\mathrm{GroupOrder}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$24$

rotation + a flip generate the dihedral group of order 8

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In$\mathrm{GroupOrder}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$8$

Properties #

Rebuilding from the generators gives back the same group

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In$\mathrm{PermutationGroup}(\mathrm{GroupGenerators}(\mathrm{AlternatingGroup}(9)))$Out$\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3, 4, 5, 6, 7, 8, 9\bigr\rbrack\bigr\rbrack)\bigr\rbrack)$

About

Examples	6
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Crosswalk	Wolfram Language

See also: Cycles, GroupOrder, GroupElements, GroupGenerators, DihedralGroup, CyclicGroup