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Klein four-group

, 00:04, 17 August 2009
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There are two equivalence classes of elements upto automorphism: the identity element as a singleton, and all the non-identity elements. All the non-identity elements are equivalent under automorphism.

==Arithmetic functions==

{| class="wikitable" border="1"
! Function !! Value !! Explanation
|-
| [[Order of a group|order]] || [[arithmetic function value::order of a group;4|4]] ||
|-
| [[Exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;2|2]] || Cyclic subgroup of order two.
|-
| [[nilpotency class]] || [[arithmetic function value::nilpotency class;1|1]] || The group is abelian.
|-
| [[derived length]] || [[arithmetic function value::derived length;1|1]] || The group is abelian.
|-
| [[Frattini length]] || [[arithmetic function value::Frattini length;1|1]] || The group is elementary abelian.
|-
| [[Fitting length]] || [[arithmetic function value::Fitting length;1|1]] || The group is abelian, hence nilpotent.
|-
| [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;2|2]] || Elementary abelian of rank two.
|-
| [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;2|2]] ||
|-
| [[max-length of a group|max-length]] || [[arithmetic function value::max-length of a group;2|2]] ||
|-
| [[Rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;2|2]] ||
|-
| [[Normal rank of a p-group|normal rank]] || [[arithmetic function value::normal rank of a p-group;2|2]] ||
|-
| [[characteristic rank of a p-group]] || [[arithmetic function value::characteristic rank of a p-group;2|2]] ||
|}
==Group properties==
{{not cyclic}}| class="wikitable" border="1"!Property !! Satisfied !! Explanation !! Comment{{|-|[[Satisfies property::Abelian}group]] || Yes || |||-|[[Satisfies property::Nilpotent group]] || Yes || |||-|[[Satisfies property::Elementary abelian group]] || Yes || |||-|[[Satisfies property::Solvable group]] || Yes || |||-|[[Satisfies property::Supersolvable group]]|| Yes || |||-|[[Dissatisfies property::Cyclic group]] || No || |||}
==Endomorphisms==