A finite group $G$ is Lagrangian (or, a CLT-group) if it satisfies the converse of Lagrange's Theorem in the sense that it has a subgroup of order equal to every divisor of its order.

Every finite nilpotent group is Lagrangian, and a finite group is supersoluble if, and only if, each of its subgroups is Lagrangian. (Finite nilpotent groups have a much stronger property: a finite group is nilpotent if, and only if, it has a normal subgroup of order $d$, for each divisor $d$ of its order.)

The class of Lagrangian groups is neither subgroup- nor quotient-closed.

The IsLagrangian( G ) command attempts to determine whether the group G is Lagrangian.  It returns true if G is Lagrangian and returns false otherwise.

A GCLT-group is a finite group $G$ such that, for each subgroup $H$ of $G$, and for each prime divisor $p$ of the index [G:H] of $H$ in $G$, there is a subgroup $L$ of $G$, containing $H$, for which the index [L:H] is equal to $p$. GCLT-groups are most commonly referred to as $𝒥$-groups in the literature.

Every GCLT-group is Lagrangian, but not conversely.

The IsGCLTGroup( G ) command attempts to determine whether the group G is a GCLT-group. It returns true if G is a GCLT-group, and returns the value false otherwise.

The group G must be an instance of a permutation group.

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

$\mathrm{IsLagrangian}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsLagrangian}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsGCLTGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsGCLTGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$

$\mathrm{true}$

$G≔\mathrm{PermutationGroup}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{Symm}\left(3\right)\right)\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(4\,5\right)\,\left(4\,5\,6\right)⟩$

$\mathrm{IsLagrangian}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsGCLTGroup}\left(G\right)$

$\mathrm{false}$

The GroupTheory[IsLagrangian] and GroupTheory[IsGCLTGroup] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.