The elements of a pc group G are always represented as words in normal form with respect to the family pcgs of G. Thus it is straightforward to compare elements of pc group, since this boils down to a mere comparison of exponent vectors with respect to the family pcgs. In particular, the word problem is efficiently solvable in pc groups.
= pcword
< pcword
However, multiplication and inversion of elements in pc groups is not as straightforward as in arbitrary finitely presented groups where a simple concatenation or reversion of the corresponding words is sufficient (but one cannot solve the word problem).
To multiply to elements in a pc group, we
first concatenate the corresponding words and then use an algorithm
called collection to transform the new word into a word in normal
form.
gap> g := FamilyPcgs( SmallGroup( 24, 12 ) ); [ f1, f2, f3, f4 ] gap> g[4] * g[1]; f1*f3 gap> (g[2] * g[3])^-1; f2^2*f3*f4
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GAP 4 manual