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):
Multiplication table of the trivial group:
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.
This is the multiplication table of a subgroup of
:
Therefore it can be extracted as a subtable of the table of
:
A group is Abelian if and only if its multiplication table is symmetric under transposition. Take the symmetric group of degree 3:
The group
is not Abelian:
When all elements of a group are involutions, the group is Abelian. That is, if the multiplication table has only 1s in the diagonal, then it is symmetric:
The multiplication table can be obtained by direct use of
PermutationProduct and
GroupElementPosition:
Two groups are isomorphic as abstract groups if they have the same group multiplication table, modulo reordering of their elements:
However, those two groups are not isomorphic as permutation groups, because their permutations have different cyclic structures:
matrix mat of integers such that element 
gives the result of the multiplication of elements i and j in the group. Positions i, j, and 
are computed with the function GroupElementPosition.
:
EXAMPLES
Basic Examples 
Scope 