Cyclic group:Z2

Verbal definition
The cyclic group of order 2 is defined as the unique group of order two. Explicitly it can be described as a group with two elements, say $$e$$ and $$x$$ such that $$ex = xe = x$$ and $$e^2 = x^2 = e$$. It can also be viewed as:


 * The quotient group of the group of integers by the subgroup of even integers
 * The multiplicative group comprising $$1$$ and $$-1$$ (in this context it is also termed the sign group)
 * The additive group of the field of two elements.
 * Bits under the XOR operation
 * The on two elements. In particular, it is a member of family::symmetric group on finite set and member of family::symmetric group of prime degree.
 * The $$GL(1,3)$$ (or equivalently, the multiplicative group of the field of order 3)
 * The multiplicative group of the ring $$\mathbb{Z}/4\mathbb{Z}$$
 * The group of units in $$\mathbb{Z}$$
 * The group $$S^0(\R)$$ viz., the group of vectors in $$\R^1$$ of unit length

This group is denoted as $$C_2$$, $$\mathbb{Z}_2$$ and sometimes as $$\mathbb{Z}/2\mathbb{Z}$$.

Multiplication table
 

Automorphisms
The cyclic group of order two has no nontrivial automorphisms. In fact, it is the only nontrivial group with the property of having no nontrivial automorphisms.

Endomorphisms
The cyclic group of order two admits two endomorphisms: the identity map and the trivial map (the map sending both elements to the identity element).

Subgroups
There are only two subgroups: the trivial subgroup and the whole group. Most of the nice subgroup properties are true for both.

Quotients
There are only two quotients: itself and the trivial quotient.

Occurrence as a subgroup
The cyclic group of order 2 occurs as a subgroup in many groups. In general, any group of even order contains a cyclic subgroup of order 2 (this follows from Cauchy's theorem, which is a corollary of Sylow's theorem, though it can also be proved by a direct counting argument). Elements of order 2, which are generators for cyclic groups of order 2, are termed involutions.

Occurrence as a normal subgroup
The cyclic group of order 2 may occur as a normal subgroup in some groups. Examples are the general linear group or special linear group over a field whose characteristic is not 2. This is the group comprising the identity and negative identity matrix.

It is also true that a normal subgroup of order two is central. More generally, it is true that a normal subgroup whose order is the least prime divisor of the order of the group is central. Thus, the existence of normal subgroups of order two indicates a nontrivial center.

Viewed another way, given a group $$G$$, it often happens that we can find a group $$H$$ with a surjective homomorphism to $$G$$ whose kernel is a cyclic group of order 2. In some cases, the Schur multiplier of the group is the cyclic group of order $$2$$. An example is the projective special linear group, whose Schur multiplier has order two and the universal central extension is the special linear group.

Occurrence as a quotient group
The cyclic group of order 2 occurs very often as a quotient. Put another way, given a group, we can often find a subgroup of index two. Any subgroup of index two is normal (more generally, any subgroup of least prime index is normal).

In these cases, the group of order two may or may not occur as a complement to the normal subgroup. Examples where it does occur as a complement are the alternating group in the symmetric group, or $$GL^+(n,\R)$$ in $$GL(n,\R)$$.

Viewed another way, given a group $$G$$, we may often be able to construct a group $$H$$ in which $$G$$ has index two.

Families
The cyclic group of order two lies in the family of cyclic groups, of general linear groups, and of symmetric groups.

It also lies in the family of sphere groups: namely, it is the group of unit vectors in $$\R^1$$. The other two sphere groups are $$S^0(\mathbb{C}) = S^1$$ (the circle group) and $$S^0(\mathbb{H}) = S^3$$ (the unit quaternion group).

Internal links

 * Linear representation theory of cyclic group:Z2
 * Permutation representation theory of cyclic group:Z2
 * Properly discontinuous group actions of cyclic group:Z2
 * Galois extensions for cyclic group:Z2
 * Group cohomology of cyclic group:Z2

Entertainment

 * Youtube link for Finite simple group of order two