A subset of a group is only a subgroup if, for every 'a' and 'b' then a*b and a-1 are also members of the group, for example, if we have the following group we could take a subset by removing the 'j' and 'k' rows and columns:
a*b |
b.1 | b.i | b.j | b.k |
| a.1 | 1 | i | j | k |
| a.i | i | -1 | k | -j |
| a.j | j | -k | -1 | i |
| a.k | k | j | -i | -1 |
This gives:
a*b |
b.1 | b.i |
| a.1 | 1 | i |
| a.i | i | -1 |
Which is a valid group because it only contains '1' and 'i' types. However, if we removing the '1' and 'k' rows and columns, then we don't get a valid group.
a*b |
b.1 | b.i | b.j | b.k |
| a.1 | 1 | i | j | k |
| a.i | i | -1 | k | -j |
| a.j | j | -k | -1 | i |
| a.k | k | j | -i | -1 |
This contains 'k' which is not an element of the group:
a*b |
b.i | b.j |
| a.i | -1 | k |
| a.j | -k | -1 |
Even Subalgebras
This applies to Clifford algebras where the elements are grouped into blades (scalars,vectors,bivectors,trivectors…). To take an even subalgebra we halve the number of dimensions by keeping the even blades (scalars,bivectors…) and removing the odd blades (vectors,trivectors…).
As we have seen on this page there are advantages to numbering the elements in bit order even though this mixes up the blades. If we do this its still easy to determine what each element is by the number of '1's it has. When they are ordered in this way we can see that each pair of original dimensions produces one dimension in the even subalgebra:
| original dimensions | subalgebra dimensions | new numbering |
| 000 = scalar | 000 = scalar | 00 |
| 001 = vector | ||
| 010 = vector | 011 = bivector | 01 |
| 011 = bivector | ||
| 100 = vector | 101 = bivector | 10 |
| 101 = bivector | ||
| 110 = bivector | 110 = bivector | 11 |
| 111 = trivector |
So we can see that the Cayley table for the even subalgebra still has the checkerboard pattern that is common to the algebras that we are studying.
[A] [B]= |
|
So the pattern of even subalgebras is the same as the Clifford algebras on which they are based, what changes is the sign of the terms.
Odd Subalgebras
Odd subalgebras are not closed so we don't tend to use them.
