Group theory – Week 02
During this (past) short week we mentioned some more results and definitions from finite group theory.
We started with the Dihedral group , which is the group of symmetries of a -sided polygon. It consists of rotations and reflections. The idea is to have the polygon laying on a plane and paint one side blue and the other one red (or whichever colors you prefer) in order to distinguish sides. This group has elements.
Next we came upon Lagrange’s theorem for finite groups:
- The order of a subgroup of a finite group is a divisor of the order of .
The order of a group is the number of elements. Meanwhile, the order of an element is defined as the power such that . It turns out that the set of elements
forms a group, the Cyclic group . If has as order a prime number then the order of any element in is .
We can also take direct products of two groups, defined as the set
and requiring that elements of different groups commute. The direct product of two groups is also a group. For example, the Klein group is isomorphic to .
Besides this we also have the set of all the permutations of objects. This forms a group of elements called the, guess what, permutation group . From Cayley we have a theorem concerning finite groups and permutation groups:
- Every finite group of order is a subgroup of the permutation group .
Then we proceeded to list of finite groups, but we got tired by order 8… At this order we have the Quaternions, which are kind of interesting.
We defined invariant subgroups, also known as normal, as a subgroup whose elements obey for any element in the higher group. When a subgroup is normal, its left cosets are equal to its right cosets. More definitions: a homomorphism is a map between groups that preserve the group multiplication laws. The kernel of a map is the set of all the elements in the domain that get mapped to the identity of the range. It turns out that
- The kernel of a homomorphism is a normal subgroup.
Finally we came upon quotient groups. We start with a group and an invariant subgroup . Then we form the set
This forms a group under the operation