44.2 Comparison of Elements of Finitely Presented Groups

Two elements of a finitely presented group are equal if they are equal in this group. Nevertheless they may be represented as different words in the generators. Because of the fundamental problems mentioned in the introduction to this chapter such a test may take very long and cannot be guaranteed to finish.

Problems get even worse when trying to compute a total ordering on the elements of a finitely presented group. As any ordering that is guaranteed to be reproducible in different runs of GAP or even with different groups given by syntactically equal presentations would be prohibitively expensive to implement, the ordering of elements is depending on a method chosen by GAP and not guaranteed to stay the same when repeating the construction of an FpGroup. The only guarantee given for the < ordering for such elements is that it will stay the same for one family during its lifetime. The attribute FpElmComparisonMethod is used to obtain a comparison function for a family of FpGroup elements.

If fam is the elements family of a finitely presented group this attribute returns a function smaller(left, right) that will be used to compare elements in fam.

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