FittingSubgroup - Maple Help

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GroupTheory

 FittingSubgroup
 construct the Fitting subgroup of a group

 Calling Sequence FittingSubgroup( G )

Parameters

 G - a permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right],\left[5,6\right],\left[7,8\right],\left[9,10\right],\left[11,12\right],\left[13,14\right],\left[15,16\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2,5,3\right],\left[4,7\right],\left[6,9\right],\left[8,11\right],\left[10,13,16,15\right],\left[12,14\right]\right]\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)\left({9}{,}{10}\right)\left({11}{,}{12}\right)\left({13}{,}{14}\right)\left({15}{,}{16}\right){,}\left({1}{,}{2}{,}{5}{,}{3}\right)\left({4}{,}{7}\right)\left({6}{,}{9}\right)\left({8}{,}{11}\right)\left({10}{,}{13}{,}{16}{,}{15}\right)\left({12}{,}{14}\right)⟩$ (1)
 > $F≔\mathrm{FittingSubgroup}\left(G\right)$
 ${F}{≔}{Fitt}{}\left(⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)\left({9}{,}{10}\right)\left({11}{,}{12}\right)\left({13}{,}{14}\right)\left({15}{,}{16}\right){,}\left({1}{,}{2}{,}{5}{,}{3}\right)\left({4}{,}{7}\right)\left({6}{,}{9}\right)\left({8}{,}{11}\right)\left({10}{,}{13}{,}{16}{,}{15}\right)\left({12}{,}{14}\right)⟩\right)$ (2)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${16}$ (3)
 > $F≔\mathrm{FittingSubgroup}\left(\mathrm{Alt}\left(4\right)\right)$
 ${F}{≔}{Fitt}{}\left({{\mathbf{A}}}_{{4}}\right)$ (4)
 > $\mathrm{GroupOrder}\left(F\right)$
 ${4}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{FittingSubgroup}\left(\mathrm{Alt}\left(6\right)\right)\right)$
 ${1}$ (6)

Compatibility

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