GAP:ConjugateSubgroups

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);  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);  gap> L := ConjugateSubgroups(G,H); [ , ,  ] gap> K := Intersection(L[1],L[2]);  gap> IdGroup(K); [ 8, 4 ]