GAP (PERMUT) - Chapter 7: Other Functions in the PERMUT Package

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7 Other Functions in the PERMUT Package
7.1 Functions
7.1-1 AllSubnormalSubgroups
7.1-2 PrimesDividingSize
7.1-3 SylowSubgroups
7.1-4 IsSCGroup
7.1-5 IsSylowTowerGroup
7.1-6 Permutizer
7.1-7 AllGeneratorsCyclicPGroup

### 7 Other Functions in the PERMUT Package

In this chapter we define some miscellaneous functions which have appeared in the context of permutability, or some functions which have been used for some of the functions of the package.

#### 7.1 Functions

##### 7.1-1 AllSubnormalSubgroups

`‣ AllSubnormalSubgroups`( G ) | ( attribute )

This function computes all subnormal subgroups of G. The method used to obtain this list consists in beginning with the list of all normal subgroups of G and by adding all normal subgroups of the subgroups in the list until no new subnormal subgroups appear. This computes the complete list of subgroups, not only a representative of each conjugacy class as other functions do.

    gap> g:=SymmetricGroup(4);
    Sym( [ 1 .. 4 ] )
    gap> AllSubnormalSubgroups(g);
    [ Sym( [ 1 .. 4 ] ), Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,4)
      (2,3), (1,3)(2,4) ]), Group(()), Group([ (1,3)(2,4) ]), Group([ (1,2)
      (3,4) ]), Group([ (1,4)(2,3) ]) ]

##### 7.1-2 PrimesDividingSize

`‣ PrimesDividingSize`( G ) | ( attribute )

This attribute gives a list of primes dividing the size of the finite group G, without repetitions. Its code has been borrowed from the GAP manual.

    gap> g:=SymmetricGroup(4);
    Sym( [ 1 .. 4 ] )
    gap> PrimesDividingSize(g);
    [ 2, 3 ]

##### 7.1-3 SylowSubgroups

`‣ SylowSubgroups`( G ) | ( attribute )

This attribute returns a list composed by one Sylow subgroup for every prime dividing the size of G. If G is soluble, then it returns a Sylow system or Sylow basis of G by means of the function `SylowSystem` (Reference: SylowSystem) (a set composed of a Sylow subgroup for each prime dividing the order of G permuting in pairs).

    gap> g:=SymmetricGroup(4);
    Sym( [ 1 .. 4 ] )
    gap> SylowSubgroups(g);
    [ Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (1,2,3) ]) ]
    gap> s5:=SymmetricGroup(5);
    Sym( [ 1 .. 5 ] )
    gap> SylowSubgroups(s5);
    [ Group([ (1,2), (3,4), (1,3)(2,4) ]), Group([ (1,2,3) ]), Group([ (1,2,3,4,
       5) ]) ]

##### 7.1-4 IsSCGroup

`‣ IsSCGroup`( G ) | ( property )

This property is `true` if G is an SC-group, and `false` otherwise. A group G is an SC-group if all its chief factors are simple. Note that a soluble group G is an SC-group if and only if G is supersoluble. The method used to check this property uses the chief series if its is available or the group is not soluble.

Since the methods for insoluble groups might on the computation of a chief series with the function `ChiefSeries` (Reference: ChiefSeries), they might not be available if the group is not given as a permutation group.

    gap> g:=SymmetricGroup(4);
    Sym( [ 1 .. 4 ] )
    gap> IsSCGroup(g);
    false
    gap> g:=GL(2,5);
    GL(2,5)
    gap> IsSCGroup(g);
    true

##### 7.1-5 IsSylowTowerGroup

`‣ IsSylowTowerGroup`( G ) | ( property )

This property takes the value `true` if \(G\) has a Sylow tower of supersoluble type, and `false` otherwise.

    gap> g:=SymmetricGroup(4);
    Sym( [ 1 .. 4 ] )
    gap> IsSylowTowerGroup(g);
    false
    gap> g:=SmallGroup(75,1);
    <pc group of size 75 with 3 generators>
    gap> IsSylowTowerGroup(g);
    true

##### 7.1-6 Permutizer

`‣ Permutizer`( G, U ) | ( function )

`‣ Permutiser`( G, U ) | ( function )

The permutiser of a subgroup U of a group G is the subgroup generated by all cyclic subgroups of G which permute with U. If U is permutable in G (in particular, if U is normal in G), then its permutizer coincides with G.

    gap> g:=SymmetricGroup(4);
    Sym( [ 1 .. 4 ] )
    gap> Permutizer(g,Subgroup(g,[(1,2,3)]));
    Group([ (1,2,3), (2,3) ])
    gap> Size(last);
    6

##### 7.1-7 AllGeneratorsCyclicPGroup

`‣ AllGeneratorsCyclicPGroup`( g, p ) | ( function )

This auxiliary function returns the list of all generators of the cyclic \(p\)-group generated by the \(p\)-element \(g\). Here \(p\) is a prime number. Since this function is not intended to be used in interactive mode, no check is done that the argument is a \(p\)-element.

    gap> AllGeneratorsCyclicPGroup((1,2,3,4,5,6,7,8,9),3);
    [ (1,2,3,4,5,6,7,8,9), (1,3,5,7,9,2,4,6,8), (1,5,9,4,8,3,7,2,6),
      (1,6,2,7,3,8,4,9,5), (1,8,6,4,2,9,7,5,3), (1,9,8,7,6,5,4,3,2) ]

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