5 The Cyclic Split Extension Method

Sections

This is a method to construct up to isomorphism the groups of order pⁿ cdotq 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 package AutPGrp. Thus it is useful to install and load this share package before using the functions described in this chapter.

5.1 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 pⁿ 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 pⁿ cdotq 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( 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> ] )

Note that the function CyclicSplitExtensionMethod requires that the groups of order pⁿ are given within the SmallGroups Library.

5.2 The Underlying Functions

It is possible to construct the cyclic extensions of a single group of order pⁿ 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 pⁿ 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 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 ) ]

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grpconst manual
January 2024