Z2 (group)
The cyclic group of degree 2 ( or ) is the smallest nontrivial group in group theory and thus the smallest finite simple group . It is isomorphic to the symmetrical group , to the first dihedral group and to the orthogonal group in the one-dimensional.
properties
Since the group is Abelian , the combination is often written additively with 0 as the neutral element and 1 as the second element of the group. This notation is suggested by its origin as a factor group of the additive group of whole numbers . The link table for this group is:
0 | 1 | |
---|---|---|
0 | 0 | 1 |
1 | 1 | 0 |
The operation of this group can be interpreted in many ways, for example as an XOR operation . A multiplicative view results from the fact that the group of invertible elements of the finite field is isomorphic to , one obtains the following multiplicative connection table, where 1 is the neutral element:
1 | 2 | |
---|---|---|
1 | 1 | 2 |
2 | 2 | 1 |
Another implementation is obtained as a unit group of the ring . This is and you get the link table
1 | −1 | |
---|---|---|
1 | 1 | −1 |
−1 | −1 | 1 |
The degree 2 cyclic group is the only order 2 group.
ℤ 2 as a subgroup
- The direct product of the cyclic group of degree 2 with itself yields the Klein four-group : .
- The direct product of countably many of these groups gives the Cantor group .
- The symmetric group contains three true subgroups that are isomorphic to the group .
Representations
Every nontrivial representation of the maps the nontrivial element to an involution , conversely every linear involution defines a representation of the .
In the case of real vector spaces, every linear involution is a reflection , so the representations of the correspond to the reflections on sub-vector spaces of any dimension.
ℤ 2 as a body
The group with the connection + given above is the additive group of a body . The multiplication required for this is through the link table
0 | 1 | |
---|---|---|
0 | 0 | 0 |
1 | 0 | 1 |
given. Note that this multiplication does not form a group. The two links and together make a body, which is often called or after the English word field for body .