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Frequently Asked QuestionsGeneral Obtaining GAP Installation Hardware/OS Usage Complaints Computing with GAP Programming GAP 7. Computing with GAP7.5: I have a finite presentation for a finite polycyclic group. When I tried to use PcGroupFpGroup() to convert it to a pcgroup, I received an error message.PcGroupFpGroup() is mainly a conversion routine. The given finite presentation must be a polycyclic presentation for PcGroupFpGroup() to work. PcGroupFpGroup() does not compute a polycyclic presentation for a finite soluble group given by an arbitrary finite presentation.See Constructing Pc Groups. In a polycyclic presentation the generators must occur in the particular order of a 'polycyclic generating sequence': As described in the Reference Manual chapter Polycyclic Groups a group G is polycyclic if there exists a subnormal series G = C_{1}> C_{2}> ... > C_{n}> C_{n+1} = {1} with cyclic factors, and a polycyclic generating sequence of G is a sequence P : = (g_{1},..., g_{n}) of elements of G such that C_{i} = <C_{i+1}, g_{i}> for 1 ≤ i ≤ n. Now for the given presentation the order of the generators is defined by the order in which the generators occur in the FreeGroup() command. This is important because it fixes the way in which commutator or conjugate relations have to be defined. Relations have to be of the form g^m / w Comm( h,g ) / w h^g / w where
Note that you may have to use brackets around w if it is a product of several generators because GAP evaluates the expressions above from left to right. Changing the order of the generators or interchanging g and h in the left hand side of a commutator or conjugate relation can produce an error message as shown by the following three examples of presentations for the symmetric group S_{3}. Example 1gap> f := FreeGroup(IsSyllableWordsFamily, "a","b"); <free group on the generators [ a, b ]> gap> a := f.1;; b := f.2;; gap> rels := [a^2,b^3,b^a/b^2]; [ a^2, b^3, a^1*b*a*b^2 ] gap> g:=f/rels; <fp group on the generators [ a, b ]> gap> Size(g); 6 gap> gpol := PcGroupFpGroup(g); <pc group of size 6 with 2 generators> Here we had started from a presentation that obeys the rules for a polycyclic presentation. Example 2gap> f := FreeGroup(IsSyllableWordsFamily, "b","a"); <free group on the generators [ b, a ]> gap> b := f.1;; a := f.2;; gap> rels := [a^2,b^3,b^a/b^2]; [ a^2, b^3, a^1*b*a*b^2 ] gap> g:=f/rels; <fp group on the generators [ b, a ]> gap> Size(g); 6 gap> gpol := PcGroupFpGroup(g); Error, illegal relator a^1*b*a*b^2 called from ... brk> Here we have just interchanged the sequence of the generators, thus breaking the rules, so that PcGroupFpGroup() does not work. Example 3gap> f := FreeGroup(IsSyllableWordsFamily, "a","b"); <free group on the generators [ a, b ]> gap> a := f.1;; b := f.2;; gap> rels := [a^2,b^3,a^b/(b*a)]; [ a^2, b^3, b^1*a*b*a^1*b^1 ] gap> g:=f/rels; <fp group on the generators [ a, b ]> gap> Size(g); 6 gap> gpol := PcGroupFpGroup(g); Error, relator b^1*a*b*a^1*b^1 is not trivial called from ... brk> Here the sequence of generators is correct, but the relator is not in the prescribed form. Some remarks


The GAP Group Last updated: Thu Jun 14 14:56:33 2012 