D8 (presented as < a,b | a4=1=b2, abab=1 >) has C2 (with elements {1-1}) as a normal subgroup. The quotient group is C22. Therefore D8 is an extension of C2 by C22. This page shows how.
The elements of the normal subgroup N are {1,-1}. Call the elements of the quotient group H {1,p,q,r} so that pq=r, pr=q, qr=p. The extension is defined by a map from H×H to N. This map is specified by colouring the Cayley table of H like this:
| * | 1 | p | q | r |
|---|---|---|---|---|
| 1 | 1 | p | q | r |
| p | p | 1 | r | q |
| q | q | r | 1 | p |
| r | r | q | p | 1 |
The pink cells of this table generate the element -1, which goes into the normal subgroup.
The correspondence between the combination {1,-1},{1,p,q,r} and the more familiar {1,a,a2,a3, b,ab,a2b,a3b} is
| 1,1 | 1 |
| -1,1 | a2 |
| 1,p | b |
| -1,p | a2b |
| 1,q | ab |
| -1,q | a3b |
| 1,r | a |
| -1,r | a3 |
|
|
|---|
| C2↑C22 |
This is a sub-page of Groups of order 8, regarded as Extensions
which describes various kinds of group extensions.
See also my main index page for groups.
Copyright N.S.Wedd 2008