Descriptions of groups of prime-power order
The authors describe an algorithm to construct a power-commutator presentation of the \(p\)-covering group \(G^*:=F/[F,R]R^p\) of a \(p\)-group \(G\) given by means of a power-commutator presentation where \(G\) can be realized as a factor \(F/R\) of the free group \(F\) with \(d\) generators and \(d\) minimal with this property. The covering algorithm is split into two major parts: (a) a tails routine extending the consistent power-commutator presentation of \(G\) to a non-consistent power-commutator presentation of the covering group by adding further defining relations and generators and appending tails of words in the new generators to the group relations, (b) a consistency routine which removes redundant generators to make the power-commutator presentation of the covering group consistent again. The article contains an old version of a tails routine not published elsewhere which -- for example -- constructs the covering group of \(B(4,4)\), the free group of exponent 4 on four generators, with 61,780 new generators. A new tails routine is introduced and proven correct which only adds 1,058 new generators. A minimum of 856 new generators is required to define the covering group. -- The new tails routine is also included in pseudo-code.
consistent power-commutator presentations
scientific article; zbMATH DE number 1204293