GAP:IsFullinvariant

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This article is about a GAP function.

This GAP function takes as input a pair of a group and a subgroup thereof. See other GAP functions like this.
This GAP function outputs a Boolean variable, i.e., it returns either true or false. View other GAP functions with Boolean output

Using this GAP function requires the loading of the following package: sonata

Definition

Package requirement

This function is part of the SONATA package.

Function type

The function takes as input two arguments, both of which are groups, and outputs a boolean answer (true/false).

Behavior

  • The function returns true if the second group is a subgroup of the first, and is fully characteristic in it.
  • The function returns false if the second group is a subgroup of the first, but is not fully characteristic.
  • If the second group is not a subgroup of the first group, but they are both subgroups of a common big group, the function returns a usage error.
  • If the two arguments are not groups, or if they are not contained in a common parent group, the function returns a NoMethodFound error.

Examples of usage

gap> G := SmallGroup(81,7);
<pc group of size 81 with 4 generators>
gap> H := Center(G);
Group([ f4 ])
gap> IsFullinvariant(G,H);
true
gap> IsFullinvariant(H,G);
Error, usage: Images(<map>), Images(<map>,<elm>), Images(<map>,<coll>) called from
Images( m, U ) called from
func( elm ) called from
ForAll( maps, function ( m )
      return IsSubset( U, Images( m, U ) );
  end ) called from
IsInvariantUnderMaps( G, U, Endomorphisms( G ) ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> IsFullinvariant(CyclicGroup(3),SymmetricGroup(3));
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `IsFullinvariant' on 2 arguments called from
Error( "usage: Images(<map>), Images(<map>,<elm>), Images(<map>,<coll>)" ); called from
Images( m, U ) called from
func( elm ) called from
ForAll( maps, function ( m )
      return IsSubset( U, Images( m, U ) );
  end ) called from
IsInvariantUnderMaps( G, U, Endomorphisms( G ) ) called from
...
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue