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5 The Cyclic Split Extension Method

This is a method to construct up to isomorphism the groups of order p^n ⋅ 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 providing slightly lower 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 package AutPGrp. Thus it is useful to install and load this package before using the functions described in this chapter.

5.1 The Main Function

5.1-1 CyclicSplitExtensionMethod

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 a normal Sylow q-subgroup.

As in Chapter 4 all groups are described as codes. Setting uncoded to true makes the function return pc groups instead.

If one wants to construct the groups of order p^n ⋅ 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.

gap> CyclicSplitExtensionMethod( 2,2,7, true );
rec( 
  both := [ <pc group of size 28 with 3 generators>, 
      <pc group of size 28 with 3 generators> ], 
  down := [ <pc group of size 28 with 3 generators>, 
      <pc group of size 28 with 3 generators> ], up := [  ] )

gap> CyclicSplitExtensionMethod( 2,2,[3,5], true );
rec( 
  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> ], 
  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> ], 
  up := [ <pc group of size 12 with 3 generators> ] )

Note that the function CyclicSplitExtensionMethod requires that the groups of order p^n are given within the Small Groups library.

5.2 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.

5.2-1 CyclicSplitExtensions

Moreover, the computation of the record entry up and the record entry down can be separated by using the following functions.

5.2-2 CyclicSplitExtensionsUp

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.

gap> G := SmallGroup( 16, 10 );;
gap> CyclicSplitExtensionsUp( G, 3, true );
[ <pc group of size 48 with 5 generators> ]

gap> G := SylowSubgroup( SymmetricGroup(4), 2);
Group([ (1,2), (3,4), (1,3)(2,4) ])
gap> CyclicSplitExtensionsDown( G, 3 );
[ rec( code := 119814529, order := 24 ), 
  rec( code := 2837724033, order := 24 ) ]

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