8.13 IsSemiRegular

IsSemiRegular( G, D )
IsSemiRegular( G, D, operation )

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

A group G operates semiregularly on a domain D if no element of G other than the idenity leaves a point of D fixed. An equal characterisation is that the stabilizer of any point of D is trivial. Yet another characterisation is that the operation of G on D is equivalent to multiple copies of 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.

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

IsSemiRegular calls
G.operations.IsSemiRegular( 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.IsSemiRegular, which computes a permutation group P that operates on [1..Length(D)] in the same way that G operates on D (see Operation) and then checks if this permutation group operations semiregularly. This of course only works because this default function is overlaid for permutation groups (see Operations of Permutation Groups).

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