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GroupElementFromWord[group,w]

returns the element of group determined by the word w in the generators of group.

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GroupElementFromWord

GroupElementFromWord[group,w]

returns the element of group determined by the word w in the generators of group.

Details

  • In GroupElementFromWord[group,w], the word w must be a list of nonzero integers {m1,…,mk} representing generators in the list returned by GroupGenerators[group]. A positive integer in the word represents the ^(th) generator, and a negative integer represents the inverse of the ^(th) generator.

Examples

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Basic Examples  (1)

For a given list of group generators, this word represents the product of the first generator with itself and then with inverse of the second generator:

Wolfram Language code: word = {1, 1, -2}

For the dihedral group of order 16, the word corresponds to the element:

Wolfram Language code: GroupElementFromWord[DihedralGroup[8], word]

The same result can be obtained by multiplying the generators explicitly:

Wolfram Language code: {gen1, gen2} = GroupGenerators[DihedralGroup[8]]
Wolfram Language code: PermutationProduct[gen1, gen1, InversePermutation[gen2]]

Scope  (1)

Reconstruct a permutation from its word in a list of generators:

Wolfram Language code: gen1 = Cycles[{{1, 4}}]; gen2 = Cycles[{{1, 2, 3, 4}}]; G = PermutationGroup[{gen1, gen2}]
Wolfram Language code: GroupElementFromWord[G, {1, 2, -1, -2}]

If the generators are given in permutation list form, then the result is also in the same form:

Wolfram Language code: G = PermutationGroup[PermutationList /@ {gen1, gen2}]
Wolfram Language code: GroupElementFromWord[G, {1, 2, -1, -2}]

Properties & Relations  (3)

GroupElementToWord constructs the word for a given group element:

Wolfram Language code: G = DihedralGroup[8]; g = Cycles[{{1, 6, 3, 8, 5, 2, 7, 4}}]; word = GroupElementToWord[G, g]

The group element can then be reconstructed with GroupElementFromWord:

Wolfram Language code: GroupElementFromWord[G, word]
Wolfram Language code: % === g

The empty word always corresponds to the identity element for any group:

Wolfram Language code: GroupElementFromWord[DihedralGroup[8], {}]
Wolfram Language code: GroupElementFromWord[AlternatingGroup[5], {}]

GroupElementFromWord is equivalent to the following function:

Wolfram Language code: groupelementfromword[group_, w_] := With[{gens = GroupGenerators[group]}, Apply[PermutationProduct, Join[gens, InversePermutation /@ Reverse[gens]][[w]]] ]
Wolfram Language code: GroupElementFromWord[AlternatingGroup[5], {1, 1, -2, 1}] === groupelementfromword[AlternatingGroup[5], {1, 1, -2, 1}]
Wolfram Research (2012), GroupElementFromWord, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupElementFromWord.html.

Text

Wolfram Research (2012), GroupElementFromWord, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupElementFromWord.html.

CMS

Wolfram Language. 2012. "GroupElementFromWord." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupElementFromWord.html.

APA

Wolfram Language. (2012). GroupElementFromWord. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupElementFromWord.html

BibTeX

@misc{reference.wolfram_2026_groupelementfromword, author="Wolfram Research", title="{GroupElementFromWord}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/GroupElementFromWord.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_groupelementfromword, organization={Wolfram Research}, title={GroupElementFromWord}, year={2012}, url={https://reference.wolfram.com/language/ref/GroupElementFromWord.html}, note=[Accessed: 12-June-2026]}