Group theory 101.02020-01-18 (2020-01-09)
This is an introduction to group theory, covering the concepts of groups, subgroups and (homo)morphisms.
Definition 1: For a set and an operation, is a group if
- There exists an element such that for all , .
- For all , there exists an element such that .
- For all , .
We usually denote by .
Exercise 1: Find/construct groups with 1, 2, 3 and 4 elements.
Exercise 2: Show that the identity element is unique. (Show that if and are identity elements, )
Exercise 3: Show that every element has a unique inverse.
Definition 2: For a group , a subset is a subgroup if
- The identity element of is contained in .
- is closed under : For all , .
- is closed under inversion: For all , .
Exercise 4: Which subgroups do the groups from exercise 1 have?
From now on, we will usually denote the group operation like multiplication: and we will just denote groups with for example instead of .
Definition 3: For groups and , a function is a (homo)morphism if for all
Definition 4: A homomorphism is an isomorphism if it is injective and bijective. Then we call and isomorphic.
An isomorphism "preserves the structure of the group". If two groups they are isomorphic, they are essentially the same.
Exercise 5: Show that there is, up to isomorphism, exactly one group with one element. That is: show that every two groups with one element are isomorphic.
Exercise 6: Show that there is, up to isomorphism, one group with two elements.
Exercise 7: What about groups with 3 or 4 elements?
Exercise 8: Show that the set
together with the normal multiplication is isomorphic to .