Automorphism Groups #3

Introduction

In this post I will explain the concept of an Automorphism Group. We will make a list of the automorphism groups of all 24 groups of order less than or equal to 12 and (to our surprise) we will see that one of these groups has as much as 168 elements and that different groups can share the same automorphism group. Finally, we will make a strategy that can be of help in finding automorphism groups in general. So far we have come across direct products of groups when we studied groups of type C2 X C2 or C2 X C2 X C2. The study of automorphism groups prepares us for the study of another type of group product, the semi-direct product. ( Which I might discuss in detail in a future post. ) In this post I consider all groups to be finite. I'll try to use as much examples as I can at first and formalize later in final wrap up. Let's begin!

Generating sets

" In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. " As so very often is the case, simple things can be hard to catch in words. Let's simplify this a bit to "If [the only subgroup of G containing X = G] then [X generates G]." So it is all about -the only subgroup of G containing X-. But we don't know X! Time for examples.

List of subgroups of C3 (cyclic group of 3 elements).
1
1, a, a^2.
The sets which are only contained in C3 are: {a}, {a^2} and {a, a^2}. Note that we can reduce this list to {a}, {a^2}. ( We will do so immediately in the next examples.) Verify that C3 has two different generating sets. C3=(a)=(a^2).

List of subgroups of D3 (dihedral group of an equilateral triangle or 3-gon).
1
1, a, a^2.
1, b
1, ba
1, ba^2
1, a, a^2, b, ba, ba^2
The sets which are only contained in D3 are: {a,b}, {a,ba}, {a,ba^2}, {a^2,b}, {a^2,ba} and {a^2, ba^2}. Verify that D3 has six different generating sets.

Automorphisms

An isomorphism from a group G to a group H is a map which is surjective, injective and preserves the group operation. An automorphism is basically an isomorphism from a group to itself. Let's ilustrate this with some examples.

Let's establish an isomorphism first.

This is a CayleyTable from a group of order 8, we call the group G.


| 0 1 2 3 4 5 6 7
- | - - - - - - - -
0 | 0 1 2 3 4 5 6 7
1 | 1 0 3 2 5 4 7 6
2 | 2 3 0 1 6 7 4 5
3 | 3 2 1 0 7 6 5 4
4 | 4 5 6 7 0 1 2 3
5 | 5 4 7 6 1 0 3 2
6 | 6 7 4 5 2 3 0 1
7 | 7 6 5 4 3 2 1 0


This is another CayleyTable from a group of order 8, we call the group H.


| a b c e b**a c**a c**b c**b**a
- | - - - - - - - -
a | e b**a c**a a b c c**b**a c**b
b | b**a e c**b b a c**b**a c c**a
c | c**a c**b e c c**b**a a b b**a
e | a b c e b**a c**a c**b c**b**a
b**a | b a c**b**a b**a e c**b c**a c
c**a | c c**b**a a c**a c**b e b**a b
c**b | c**b**a c b c**b c**a b**a e a
c**b**a | c**b c**a b**a c**b**a c b a e


$\begin{array}{cccccccccc}
& | & a & b & c & e & b\text{**}a & c\text{**}a & c\text{**}b & c\text{**}b\text{**}a \\
- & | & - & - & - & - & - & - & - & - \\
a & | & e & b\text{**}a & c\text{**}a & a & b & c & c\text{**}b\text{**}a & c\text{**}b \\
b & | & b\text{**}a & e & c\text{**}b & b & a & c\text{**}b\text{**}a & c & c\text{**}a \\
c & | & c\text{**}a & c\text{**}b & e & c & c\text{**}b\text{**}a & a & b & b\text{**}a \\
e & | & a & b & c & e & b\text{**}a & c\text{**}a & c\text{**}b & c\text{**}b\text{**}a \\
b\text{**}a & | & b & a & c\text{**}b\text{**}a & b\text{**}a & e & c\text{**}b & c\text{**}a & c \\
c\text{**}a & | & c & c\text{**}b\text{**}a & a & c\text{**}a & c\text{**}b & e & b\text{**}a & b \\
c\text{**}b & | & c\text{**}b\text{**}a & c & b & c\text{**}b & c\text{**}a & b\text{**}a & e & a \\
c\text{**}b\text{**}a & | & c\text{**}b & c\text{**}a & b\text{**}a & c\text{**}b\text{**}a & c & b & a & e
\end{array}$

( Got the table in TeX but as you can see, blogger format is too small... )

We will investigate if G,H are isomorphic and ( if so ) then define an isomorphism f: G-> H.
( Post in progress, thus more later... )