attempt to determine whether a group is Lagrangian
determine whether a group is a GCLT group
IsLagrangian( G )
IsGCLTGroup( G )
a finite group
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.
The following examples illustrate that the class of Lagrangian groups is not subgroup-closed.
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 ≔ PermutationGroup⁡DirectProduct⁡CyclicGroup⁡3,Symm⁡3
The GroupTheory[IsLagrangian] and GroupTheory[IsGCLTGroup] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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