42.5 Elementary Operations for a Pcgs and an Element

The relative order of elm with respect to the prime order pcgs pcgs.

returns the pos-th exponent of elm with respect to pcgs.

returns the exponents of elm with respect to pcgs. The second form returns the exponents in the positions given in posran.

returns the depth of the element elm with respect to pcgs.

returns the leading exponent of elm with respect to pcgs.

returns the element corresponding to the exponent vector list with respect to pcgs. The exponents in list must be in the range of permissible exponents for pcgs. It is not guaranteed that PcElementByExponents will reduce the exponents modulo the relative orders. (You should use the operation LinearCombinationPcgs for this purpose.) The NC version does not check that the lengths of the lists fit together and does not check the exponent range.

The third version gives exponents only wrt. the generators in pcgs indexed by basisind.

returns the product Õ_ipcgs [i]^{list [i]}. In contrast to PcElementByExponents this permits negative exponents.

gap> G := Group( (1,2,3,4),(1,2) );; P := Pcgs(G);;
gap> g := PcElementByExponents(P, [0,1,1,1]);
(1,3,4)
gap> ExponentsOfPcElement(P, g);
[ 0, 1, 1, 1 ]

sifts elm through pcgs, reducing it if the depth is the same as the depth of one of the generators in pcgs. Thus the identity is returned if elm lies in the group generated by pcgs. pcgs must be an induced pcgs and elm must lie in the span of the parent of pcgs.

reduces elm at the induces pcgs ipcgs such that the exponents of the reduced result r are zero at the depths for which there are generators in ipcgs. Elements, whose quotient lies in the group generated by ipcgs yield the same canonical element.

reduces the element x by dividing off (from the left) a power of y such that the leading coefficient of the result with repect to pcgs becomes zero. The elements x and y therefore have to have the same depth.

returns an element in the span of pcgs whose exponents for indices 1 to dep -1 with respect to pcgs are the same as those of elm, the remaining exponents are undefined. This can be used to obtain more ``simple'' elements if only representatives in a factor are required, see Factor Groups of Polycyclic Groups - Modulo Pcgs.

The difference to HeadPcElementByNumber (see HeadPcElementByNumber) is that HeadPcElementByNumber is guaranteed to zero out trailing coefficients while CleantedTailPcElement will only do this if it can be done cheaply.

returns an element in the span of pcgs whose exponents for indices 1 to dep -1 with respect to pcgs are the same as those of elm, the remaining exponents are zero. This can be used to obtain more ``simple'' elements if only representatives in a factor are required.

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