Action of Inner Automorphisms on Subgroup Lattices

This Demonstration illustrates the action of an inner automorphism of a group on the subgroup lattice of . The subgroup lattice is shown in blue and the action is shown in red. Move the cursor over a subgroup to display its description. Five non-Abelian groups of small order are included as examples, and any element in the selected group can be used to induce the automorphism.

A group homomorphism is a mapping from a group to a group such that for all and in . An automorphism of a group is a one-to-one onto homomorphism from a group to itself. If is an element in a group , the mapping for all in is an inner automorphism of , called conjugation by . Since all normal subgroups are fixed by an inner automorphism, no examples of Abelian groups are included, nor is the quaternion group of order 8.

The elements of and are given in cycle notation; the elements in the dihedral groups are given in terms of symmetries of the corresponding -gon.