Strictly speaking, GAP 3 did not support elements of finitely presented groups. Instead, the ``words in abstract generators'' of the underlying free groups were (mis)used. This caused problems whenever calculations with elements were involved, the most obvious ones being wrong results of element comparisons. Also functions that should in principle work for any group were not applicable to finitely presented groups. In effect, a finitely presented group had to be treated in a special way in GAP 3.
GAP 4 distinguishes free groups and their elements from finitely presented groups and their elements. Comparing two elements of a finitely presented group will yield either the correct result or no result at all.
Note that in GAP 4, the arithmetic and comparison operations for
group elements do not depend on a context provided by a group that
contains the elements. In particular, in GAP 4 it is not
meaningful to call Order( G, g ) for a group G and an element
g.
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GAP 4 manual