gives the multiplication table of group as an array.
Examplesopen allclose all
Basic Examples (1)
These are the multiplications of all pairs of elements, numbered as returned by GroupElementPosition:
In permutation-group algebra the basic elements are linear combinations of the permutations of a group. It is possible to avoid recomputation of products of permutations by using a multiplication table. Denoting elements of the group algebra as lists of coefficients, it is possible to multiply them:
Properties & Relations (7)
Every row and every column of the multiplication table of a group contains every permutation once, but in different order. Hence, the table is a Latin square (note that not every Latin square corresponds to a group, because associativity is not guaranteed):
The Cayley theorem states that every finite group is isomorphic to a subgroup of some symmetric group of permutations. Hence every multiplication table is a subtable of the table of a symmetric group, perhaps after renumbering of permutations.
Possible Issues (1)
Wolfram Research (2010), GroupMultiplicationTable, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupMultiplicationTable.html.
Wolfram Language. 2010. "GroupMultiplicationTable." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupMultiplicationTable.html.
Wolfram Language. (2010). GroupMultiplicationTable. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupMultiplicationTable.html