Action of Inner Automorphisms on Subgroup Lattices
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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.
Contributed by: Marc Brodie (July 2020)
(Benedictine University Mesa)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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.
Permanent Citation
Applying the Pólya-Burnside Enumeration Theorem
Hector Zenil and Oleksandr Pavlyk
A Cycle Index Spreadsheet
Seth J. Chandler
Social Golfer Problem
Ed Pegg Jr
Subgroup Lattices of Finite Cyclic Groups
Marc Brodie
Subgroup Lattices of Groups of Small Order
Marc Brodie
Lattice of Subgroups of Permutation Groups
Jaime Rangel-Mondragon
Matrix Representation of the Permutation Group
Jaime Rangel-Mondragon
Golay Code
Ed Pegg Jr
Cycles from Permutations
Seth J. Chandler
Lengths of Permutation Cycles
Michael Schreiber