IsGCLTGroup - Maple Help

GroupTheory

 IsLagrangian
 attempt to determine whether a group is Lagrangian
 IsGCLTGroup
 determine whether a group is a GCLT group

 Calling Sequence IsLagrangian( G ) IsGCLTGroup( G )

Parameters

 G - a finite group

Description

 • 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.

Examples

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

The following examples illustrate that the class of Lagrangian groups is not subgroup-closed.

 > $\mathrm{IsLagrangian}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsLagrangian}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsGCLTGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{IsGCLTGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$
 ${\mathrm{true}}$ (4)

The smallest Lagrangian group that is not a GCLT-group is the direct product of a cyclic group of order $3$ and the symmetric group of degree $3$.

 > $G≔\mathrm{PermutationGroup}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{Symm}\left(3\right)\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({4}{,}{5}\right){,}\left({4}{,}{5}{,}{6}\right)⟩$ (5)
 > $\mathrm{IsLagrangian}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsGCLTGroup}\left(G\right)$
 ${\mathrm{false}}$ (7)

Compatibility

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