Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution - Noncommercial - No Derivative Works 3.0 United States License.
By Cayley's Theorem, all groups can be represented as permutation groups, subgroups of the full permutation group on the group's domain. For finite groups, the set on which the permutations are performed can be simplifed to a set of integers with cardinality equal to the order of the group.
These topics are introduced in Chapter 15 of Applied Discrete Structures.
Individual elements of are represented with cycle notation and the wrapper
Use an asterisk,
We can make a similar observation when we conjugate
Here is a list of powers of
Of course since
Other common groups are available, represented with permutations. Here we examine the dihedral group . Among the other groups are cyclic groups (CyclicPermutationGroup); dicyclic groups (DiCyclicGroup), which include the quaternion group; and the alternating groups (AlternatingGroup).
Knowing a set of generators for , we could have used
For each subgroup of , we print the order of the subgroup followed by a generating set. This shows that all subgroups of are cyclic but itself is not.
Here are the normal subgroups of .
Here, we see why the subgroup generated by isn't normal because we get a different subgroup when whe conjugate it with .