The Fitting subgroup of a finite group $G$ is the unique largest normal nilpotent subgroup of $G$. Its existence and uniqueness is guaranteed by Fitting's Theorem, which asserts that the product of a family of normal and nilpotent subgroups of a finite group $G$ is again a normal and nilpotent subgroup of $G$.

The Fitting subgroup of $G$ is also equal to the (direct) product of the $p$-cores of $G$, as $p$ ranges over the prime divisors of the order of $G$.

If $G$ is a soluble group, then the Fitting subgroup of $G$ is nontrivial.

The FittingSubgroup( G ) command constructs the Fitting subgroup  of a group G. The group G must be an instance of a permutation group.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\,\left[5\,6\right]\,\left[7\,8\right]\,\left[9\,10\right]\,\left[11\,12\right]\,\left[13\,14\right]\,\left[15\,16\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\,5\,3\right]\,\left[4\,7\right]\,\left[6\,9\right]\,\left[8\,11\right]\,\left[10\,13\,16\,15\right]\,\left[12\,14\right]\right]\right)\right)$

$G≔⟨\left(1\,2\right)\left(3\,4\right)\left(5\,6\right)\left(7\,8\right)\left(9\,10\right)\left(11\,12\right)\left(13\,14\right)\left(15\,16\right)\,\left(1\,2\,5\,3\right)\left(4\,7\right)\left(6\,9\right)\left(8\,11\right)\left(10\,13\,16\,15\right)\left(12\,14\right)⟩$

$F≔\mathrm{FittingSubgroup}\left(G\right)$

$F≔Fitt\left(⟨\left(1\,2\right)\left(3\,4\right)\left(5\,6\right)\left(7\,8\right)\left(9\,10\right)\left(11\,12\right)\left(13\,14\right)\left(15\,16\right)\,\left(1\,2\,5\,3\right)\left(4\,7\right)\left(6\,9\right)\left(8\,11\right)\left(10\,13\,16\,15\right)\left(12\,14\right)⟩\right)$

$\mathrm{GroupOrder}\left(F\right)$

$16$

$F≔\mathrm{FittingSubgroup}\left(\mathrm{Alt}\left(4\right)\right)$

$F≔Fitt\left({\mathbf{A}}_4\right)$

$\mathrm{GroupOrder}\left(F\right)$

$4$

$\mathrm{GroupOrder}\left(\mathrm{FittingSubgroup}\left(\mathrm{Alt}\left(6\right)\right)\right)$

$1$

The GroupTheory[FittingSubgroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.