compute an elementary abelian regular normal subgroup of a primitive permutation group
EARNS( G )
PermutationGroup; a permutation group
For a permutation group G, an "EARNS" is a normal subgroup of G that is elementary abelian and acts regularly on the domain of action of G. A permutation group may, or may not, possess an EARNS.
The EARNS( G ) command returns an EARNS for a permutation group G, provided that one exists, and returns FAIL if G has no EARNS.
It is clear that for a permutation group to posess an EARNS it must be transitive and its support must have prime power cardinality. Therefore, EARNS returns FAIL if either of these conditions is not true.
In general, for Maple to identify an EARNS for a permutation group the group must either be primitive or a Frobenius group (or both). If G is neither primitive nor a Frobenius group, then EARNS may raise an exception indicating that the group is imprimitive and that Maple cannot, in that case, determine whether or not G has an EARNS.
G ≔ Symm⁡3
G ≔ Symm⁡4
E ≔ EARNS⁡G
A group acting on a set not of prime power cardinality can have no EARNS.
G ≔ CyclicGroup⁡10
An intransitive group cannot posess an EARNS.
G ≔ Group⁡Perm⁡1,2,3,4,5
G ≔ Group⁡Perm⁡2,7,4,8,6,5,3,Perm⁡2,4,3,6,8,7,Perm⁡1,2,3,4,5,6,7,8
Primitive Frobenius groups always have an EARNS, the Frobenius kernel.
G ≔ FrobeniusGroup⁡14520,2
G≔ < a permutation group on 121 letters with 5 generators >
E ≔ EARNS⁡G:
G ≔ Symm⁡2048
G ≔ Alt⁡55
A regular elementary abelian transitive group is its own EARNS, even if it does not act primitively.
G ≔ TransitiveGroup⁡27,4:
Some imprimitive Frobenius groups have an EARNS.
G ≔ TransitiveGroup⁡25,9
But not all do.
G ≔ TransitiveGroup⁡16,63
In most cases, however, an exception is raised if the input to EARNS is imprimitive.
G ≔ WreathProduct⁡CyclicGroup⁡3,CyclicGroup⁡3
Error, (in GroupTheory:-EARNS) group must be primitive
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