There are certain products of elements whose exponents are used often within algorithms, and which might be obtained more easily than by computing the product first and to obtain its exponents afterwards. The operations in this section provide a way to obtain such exponent vectors directly.
(The circumstances under which these operations give a speedup depend very much on the pcgs and the representation of elements that is used. So the following operations are not guaranteed to give a speedup in every case, however the default methods are not slower than to compute the exponents of a product and thus these operations should always be used if applicable.)
ExponentsConjugateLayer( mpcgs, elm, e ) O
Computes the exponents of elm e with respect to mpcgs. elm must be in the span of mpcgs, e an pc element in the span of the parent pcgs of mpcgs and mpcgs must be the modulo pcgs for an abelian layer. (This is the usual case when acting on a chief factor). In this case if mpcgs is induced by the family pcgs, the exponents can be computed directly by looking up exponents without having to compute in the group and having to collect a potential tail.
The second class are exponents of products of the generators which make up
the pcgs. If the pcgs used is a FamilyPcgs these exponents can be looked
up and do not need to be computed.
ExponentsOfRelativePower( pcgs, i ) O
For p = pcgs [i ] this function returns the exponent vector with respect to pcgs of the elment pe where e is the relative order of p in pcgs. For the family pcgs or pcgs induced by it, this might be faster than computing the element and computing its exponent vector.
ExponentsOfConjugate( pcgs, i, j ) O
returns the exponents of pcgs [i ]pcgs [j ] with respect to pcgs. For the family pcgs or pcgs induced by it, this might be faster than computing the element and computing its exponent vector.
ExponentsOfCommutator( pcgs, i, j ) O
returns the exponents of the commutatior Comm( pcgs [i ],pcgs [j ]) with respect to pcgs. For the family pcgs or pcgs induced by it, this might be faster than computing the element and computing its exponent vector.
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GAP 4 manual