Post #5: Finite groups, subgroups and Hasse diagrams

 An introductory comment

 

In most math classes, the student is taught methods for solving certain types of problems. Abstract Algebra is different, because it is about the rules of mathematical structures and sub-structures. Today we talk about subgroups that have their own subgroups and how to represent the chain of subgroups from the original group G down to the smallest subgroup, which is always the identity e by itself.

 

Examples of Zn and their subgroups

The additive group Zn has exactly n elements. It turns out that the subgroups will have orders of the numbers that divide n, and more than that, if m divides n, there must be a subgroup of Zn with order m.

 

Here are a few examples.

 

Z3 = ({0,1,2},+)

3 is prime, so the only subgroup is ({0},+).

 

Z4 = ({0,1,2,3},+)

4 = 2✕2, so the the subgroups are

the multiples of 2, ({0,2},+), and

({0},+).

 

Z18 = ({0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17},+)

18 = 2✕3✕3, so we have more subgroups.

The multiples of 2, ({0,2,4,6,8,10,12,14,16},+)

The multiples of 3, ({0,3,6,9,12,15},+)

The multiples of 6, ({6,12,},+)

The multiples of 9, ({0,9},+)

and ({0},+), the subgroup that is always there in any group.

 

Some subgroups contain subgroups of their own, and to show this relation, Hasse diagrams are used. The are graphs where all the subgroups are listed from top to bottom. If a subgroup is contained in a larger group or subgroup on the list, a arrow will connect them and because containment is transitive, which is to say if A⊇B and B⊇C, then A⊇C, we do not need to add arrows that show a set of high order contains a set two levels down, it is implied.

 

Here are the Hasse diagrams for the groups and subgroups described above. 

 

 Commentary

 

As you can read in his biography, Helmut Hasse's reputation is stained by his relationship to the Nazi party. He was happy to collaborate with Jewish colleagues and had some Jewish blood, but his sympathy politically was with the Nazis. He tried to join but was rejected because of his Jewish ancestry, and some hardcore Nazis objected to his promotion in the mathematics department at Göttingen when Hermann Weyl, a great Jewish mathematician and longtime collaborator with Albert Einstein, decided to join Einstein at Princeton's Institute of Advanced Studies in 1933 to escape the Nazis. In 1934, Hasse finally got the post.

 

During the war, Hasse worked for the German Navy, working on ballistics problems. In 1945, The British occupation forces denied him the right to go back to Göttingen. By 1948, he was able to be hired as a professor, and in his post-war career, taught in both East Germany and West Germany. In the late 1940s, he published a influential text on algebraic number theory that was later translated into in English.

 

Tomorrow: Our first example of a non-abelian group, where there exist elements a and b where a✕b ≠ b✕a.