WOLFRAM|DEMONSTRATIONS PROJECT

The 30 Subgroups of the Symmetric Group on Four Symbols

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subgroup
1
There are 30 subgroups. This subgroup is not abelian.
set =
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}
inverse =
{1,2,3,5,4,6,7,8,13,19,14,20,9,11,15,21,17,23,10,12,16,22,18,24}
order =
{1,2,2,3,3,2,2,2,3,4,4,3,3,4,2,3,2,4,4,3,3,2,4,2}
The symmetric group
S
n
on a finite set of
n
symbols is the group whose elements are all
n!
permutations of the
n
symbols. The group operation is the composition of such permutations, which are bijective functions from the set of symbols to itself.