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. ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle C_{2}}$ ${\displaystyle S_{2}}$ ${\displaystyle D_{1}}$ ${\displaystyle O(1)}$

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: ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} }$

${\displaystyle +}$	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: ${\displaystyle \{1,2\}}$ ${\displaystyle \mathbb {Z} _{3}=\{0,1,2\}}$ ${\displaystyle \mathbb {Z} _{2}}$

${\displaystyle \cdot }$	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 ${\displaystyle \mathbb {Z} }$ ${\displaystyle \{-1,1\}\subset \mathbb {Z} }$

${\displaystyle \cdot }$	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 : . ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}=V_{4}}$
• The direct product of countably many of these groups gives the Cantor group .
• The symmetric group ${\displaystyle S_{3}}$ contains three true subgroups that are isomorphic to the group . ${\displaystyle \mathbb {Z} _{2}}$

Representations

Every nontrivial representation of the maps the nontrivial element to an involution , conversely every linear involution defines a representation of the . ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{2}}$

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. ${\displaystyle \mathbb {Z} _{2}}$

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 ${\displaystyle \mathbb {Z} _{2}=\{0,1\}}$ ${\displaystyle \mathbb {Z} _{2}}$

${\displaystyle \cdot }$	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 . ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle +}$ ${\displaystyle \cdot }$ ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle F_{2}}$ ${\displaystyle \mathbb {F} _{2}}$

See also

• List of small groups

Web links

• Small order groups