%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W cyclic.tex GrpConst documentation Bettina Eick %% %H $Id: cyclic.tex,v 1.5 1999/11/07 19:28:45 gap Exp $ %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{The Cyclic Split Extension Method} \index{The Cyclic Split Extension Method} This is a method to construct up to isomorphism the groups of order $p^n \cdot q$ for different primes $p$ and $q$ which have a normal Sylow subgroup. We first describe the main function for this method and then functions for a slightly more low level access to the algorithms. Note that all functions described in this chapter rely on an efficient method for `AutomorphismGroup' for $p$-groups. Such a method is provided in the forthcoming share package AutPGrp. Thus it is useful to install and load this share package before using the functions described in this chapter. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{The Main Function} \> CyclicSplitExtensionMethod( <p>, <n>, <q> ) F \> CyclicSplitExtensionMethod( <p>, <n>, <q>, <uncoded> ) F Clearly, each of the computed groups is a split extension of a group of order $p^n$ and the cyclic group of order $q$. The output is a record with three entries <up>, <down> and <both>. Each of these contains a list of groups, <both> the nilpotent groups, <up> the remaining groups with a normal Sylow $p$-subgroup and <down> the remaining groups with normal Sylow $q$-subgroup. As in Chapter "The Frattini Extension Method" all groups are described as codes. Setting <uncoded> to true, the function will return pc groups instead. If one wants to construct the groups of order $p^n \cdot q$ for fixed $p$ and several primes $q$, it is more efficient to do this in one go. Thus it is possible to hand a list of primes for the input <q>. \beginexample gap> CyclicSplitExtensionMethod( 2,2,7, true ); rec( up := [ ], down := [ <pc group of size 28 with 3 generators>, <pc group of size 28 with 3 generators> ], both := [ <pc group of size 28 with 3 generators>, <pc group of size 28 with 3 generators> ] ) gap> CyclicSplitExtensionMethod( 2,2,[3,5], true ); rec( up := [ <pc group of size 12 with 3 generators> ], down := [ <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators>, <pc group of size 20 with 3 generators>, <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators> ], both := [ <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators>, <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators> ] ) \endexample Note that the function `CyclicSplitExtensionMethod' requires that the groups of order $p^n$ are given within the SmallGroups Library. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{The Underlying Functions} It is possible to construct the cyclic extensions of a single group of order $p^n$ only. The output is as above. \>CyclicSplitExtensions( <G>, <q> ) F \>CyclicSplitExtensions( <G>, <q>, <uncoded> ) F Moreover, the computation of the record entry <up> and the record entry <down> can be separated by using the following functions. \> CyclicSplitExtensionsUp( <G>, <q> ) F \> CyclicSplitExtensionsUp( <G>, <q>, <uncoded> ) F \> CyclicSplitExtensionsDown( <G>, <q> ) F \> CyclicSplitExtensionsDown( <G>, <q>, <uncoded> ) F The input for these functions is the same as above. The first function returns a list of groups with one normal subgroup of order $p^n$ and the second a list of groups with one normal subgroup of order $q$. \beginexample gap> G := SmallGroup( 16, 10 );; gap> CyclicSplitExtensionsUp( G, 3, true ); [ <pc group with 5 generators> ] gap> G := SylowSubgroup( SymmetricGroup(4), 2); Group([ (1,2), (3,4), (1,3)(2,4) ]) gap> CyclicSplitExtensionsDown( G, 3 ); [ rec( code := 6562689, order := 24 ), rec( code := 2837724033, order := 24 ) ] \endexample