8.12 IsRegular

IsRegular( G, D ) IsRegular( G, D, operation )

IsRegular returns true if the group G operates regularly on the domain D, which must be a list of points of arbitrary type, and false otherwise.

A group G operates regularly on a domain D if it operates transitively and no element of G other than the idenity leaves a point of D fixed. An equal characterisation is that G operates transitively on D and the stabilizer of any point of D is trivial. Yet another characterisation is that the operation of G on D is equivalent to the operation of G on its elements by multiplication from the right.

It is not allowed that D is a proper subset of a domain, i.e., D must be invariant under the operation of G.

IsRegular accepts a function operation of two arguments d and g as optional third argument, which specifies how the elements of G operate (see Other Operations).

    gap> g := Group( (1,2,3)(6,7), (3,4,5)(7,8) );;
    gap> IsRegular( g, [1..5] );
    false
    gap> IsRegular( g, Elements(g), OnRight );
    true
    gap> g := Group( (1,2,3), (3,4,5) );;
    gap> IsRegular( g, Orbit( g, [1,2,3], OnTuples ), OnTuples );
    true 

IsRegular calls
G.operations.IsRegular( G, D, operation )
and returns the value. Note that the third argument is not optional for functions called this way.

The default function called this way is GroupOps.IsRegular, which tests if G operates transitively and semiregularly on D (see IsTransitive and IsSemiRegular).

Previous Up Top Next
Index