The Character tables of the groups of order 8: Z_8

 ℤ8 is generated by a single element, the square root of i, which I will denote by the letter a. Wherever a is sent, all other entries will be to the appropriate power of a. The eight elements will be called 1, a, i, ai, -1, -a, -i and -ai.

 *    1    a    i   ai   -1   -a   -i   -ai

1st | 1    1    1    1    1    1    1    1 

2nd | 1    a    i   ai   -1   -a   -i   -ai

3rd | 1    i   -1   -i    1    i   -1   -i 

4th | 1   ai   -i    a   -1   -ai   i   -a 

5th | 1   -1    1   -1    1    -1    1  -1 

6th | 1   -a    i   -ai  -1     a   -i  ai

7th | 1   -i   -1   i     1    -i   -1   i

8th | 1   -ai  -i  -a    -1    ai    i   a

Checking for orthogonality is a little tougher because you must turn one row into its complex conjugates before doing the dot product, but it does still work. This orthogonality makes group representation theory a useful part of physically describing an object whose group actions are represented by some set of matrices.

 

Tomorrow: The character tables for the quaterions and D4. 

Post #10: Two groups of order 8, the abelian group Z_8 and the non-abelian group D_4.

 The abelian group Z8 is as uncomplicated as a group of order 8 can get. It has a subgroup order 4, another of order 2, and a third of order 1, which is to say the identity. Let's represent it as an additive group and make the Hasse diagram of the subgroup structure.

The structure of subgroups is like nesting Russian dolls, where the next smallest subgroup always fits inside the previous set in the structure.

Easy peasy, lemon squeezy.

The Hasse diagram for D4 is much more complex. There are three subgroups of order 4 and five subgroups of order 8. The subgroup {I, R180°} is contained in every one of the order 4 subgroups. The only element all the order 2 subgroups have in common is the identity I.

 

We have not yet explored the group structure of the subgroups of order 4 that contain two reflections. Tomorrow, we will look at the Cayley table and Hasse diagram of this abelian group known as the Klein four-group or Z2×Z2.