8.7 IsFixpoint

IsFixpoint( G, d )
IsFixpoint( G, d, operation )

IsFixpoint returns true if the point d is a fixpoint under the operation of the group G.

We say that d is a fixpoint under the operation of G if every element g of G maps d to itself. This is equivalent to saying that each generator of G maps d to itself.

As a special case it is allowed that the first argument is a single group element, though this does not make a lot of sense, since in this case IsFixpoint simply has to test d^g = d.

IsFixpoint 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> IsFixpoint( g, 1 );
    false
    gap> IsFixpoint( g, [6,7,8], OnSets );
    true 

IsFixpoint is so simple that it does all the work by itself, and, unlike the other functions described in this chapter, does not dispatch to another function.

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