This article is about a GAP function.
NormalSubgroups is a GAP command that takes in one argument representing a group and outputs a list of groups.
Applying NormalSubgroups to a given group returns a list of all its normal subgroups.
Examples of usage
Some examples involving prespecified groups
gap> NormalSubgroups(SymmetricGroup(3)); [ Group(()), Group([ (1,2,3) ]), Sym( [ 1 .. 3 ] ) ] gap> L := NormalSubgroups(SymmetricGroup(4)); [ Group(()), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), Sym( [ 1 .. 4 ] ) ] gap> K := List(L,IdGroup); [ [ 1, 1 ], [ 4, 2 ], [ 12, 3 ], [ 24, 12 ] ]
The first example lists all the normal subgroups of the symmetric group on three letters. The set here is . Each of these normal subgroups (except the whole group) is described by its generating set.
The second example computes the normal subgroups of the symmetric group on four letters and outputs the list of its normal subgroups. This list is stored with the variable name L. In the next command, the members of this list are mapped to their group IDs, using the IdGroup command. The GAP:List command is used to achieve the mapping on each member of the list.