Cyclic group:Z2

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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 $e<math>and<math>x$ such that $ex=xe=x$ and $e^{2}=x^{2}=e$. It can also be viewed as:

• The quotient 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)
• Bits under the XOR operation
• The symmetric group on two elements

Properties

Cyclicity

This particular group is cyclic

Abelianness

This particular group is Abelian

Nilpotence

This particular group is nilpotent

Solvability

This particular group is solvable

Endomorphisms

Automorphisms

The cyclic group of order two has no nontrivial automorphisms. In fact, it is the only Abelian 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.

External links

Entertainment

• Lyrics for Finite simple group of order two by M. Salomone
• Youtube link for Finite simple group of order two