GAP:ConjugateSubgroups

This article is about a GAP function.

Definition

Type

The function ConjugateSubgroups is a GAP command that takes as input two groups, typically, with the second group being a subgroup of the first. It outputs a list of groups.

Behavior

The behavior of the function is as follows:

• If both groups are subgroups of a common big group, then GAP returns the list of all conjugates of the second subgroup under the action of the first. In particular, if the second group is a subgroup of the first group, then GAP returns the list of all its conjugate subgroups.
• If the two groups are not subgroups of a common big group, then GAP returns a NoMethodFound error.

Typical use

The typical use of the function is as follows:

ConjugateSubgroups(group,subgroup);

Note that if the order of the group and subgroup is reversed, GAP will not return an error and instead simply give a one-element list comprising only the group.

Another somewhat less typical use is:

ConjugateSubgroups(subgroup1,subgroup2);

Where we want to find all conjugates of subgroup2 under the action of subgroup1.

Related functions

• GAP:IsNormal takes two groups, both of which are subgroups of a common big group, and returns whether the first group normalizes the second group. In its typical use, it tests whether a subgroup is normal in the whole group.
• GAP:ConjugacyClassesSubgroups lists all conjugacy classes of subgroups of a group.

Examples of usage

gap> ConjugateSubgroups(SymmetricGroup(3),SymmetricGroup(2));
[ Group([ (1,2) ]), Group([ (2,3) ]), Group([ (1,3) ]) ]
gap> ConjugateSubgroups(SymmetricGroup(2),SymmetricGroup(3));
[ Group([ (1,2,3), (1,2) ]) ]
gap> G := DihedralGroup(8);
<pc group of size 8 with 3 generators>
gap> C := Center(G);
Group([ f3 ])
gap> ConjugateSubgroups(G,C);
[ Group([ f3 ]) ]

The first example above lists the conjugates to the subgroup of the symmetric group on $\{1,2,3\}$ that is the symmetric group on $\{1,2\}$. The second example gives conjugates to the symmetric group on $\{1,2,3\}$ under the action of the subgroup which is the symmetric group on $\{1,2\}$ (which is a single-element set).

The remaining example starts with a group defined as the dihedral group of order eight, considers a subgroup defined as the center, and then tries to compute its conjugate subgroups.

Here are some more examples:

gap> G := GL(2,3);
GL(2,3)
gap> H := SylowSubgroup(G,2);
<group of 2x2 matrices of size 16 in characteristic 3>
gap> L := ConjugateSubgroups(G,H);
[ <group of 2x2 matrices of size 16 in characteristic 3>,
  <group of 2x2 matrices of size 16 in characteristic 3>,
  <group of 2x2 matrices of size 16 in characteristic 3> ]
gap> K := Intersection(L[1],L[2]);
<group of 2x2 matrices in characteristic 3>
gap> IdGroup(K);
[ 8, 4 ]