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


  • 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.


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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:

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

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

Scope  (1)

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

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

Properties & Relations  (3)

GroupElementToWord constructs the word for a given group element:

The group element can then be reconstructed with GroupElementFromWord:

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

GroupElementFromWord is equivalent to the following function:

Wolfram Research (2012), GroupElementFromWord, Wolfram Language function,


Wolfram Research (2012), GroupElementFromWord, Wolfram Language function,


Wolfram Language. 2012. "GroupElementFromWord." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2012). GroupElementFromWord. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_groupelementfromword, author="Wolfram Research", title="{GroupElementFromWord}", year="2012", howpublished="\url{}", note=[Accessed: 02-December-2023 ]}


@online{reference.wolfram_2023_groupelementfromword, organization={Wolfram Research}, title={GroupElementFromWord}, year={2012}, url={}, note=[Accessed: 02-December-2023 ]}