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Klein four-group

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{{particular group}}
 {{smallest|non-[[cyclic groupImportance rank::1| ]]}} {{group of order|4}} 
* It is the group comprising the elements <math>(\pm 1, \pm 1)</math> under coordinate-wise multiplication
* It is the unique non-cyclic group of order 4
* It is the subgroup of [[symmetric group:S4|the symmetric group of degre degree four]] comprising the double transpositions, and the identity element.* It is the [[member of family::Burnside group]] <math>B(2,2)</math>: the ''free group'' on two generators with exponent two.
===Multiplication table===
{| class="wikitable" border="1"! Element !! <math>e</math> !! <math>a</math> !! <math>b</math> !! <math>c</math>|-|<math>e</math> || <math>e</math> || <math>a</math> || <math>b</math> || <math>c</math>|-|<math>a</math> || <math>a</math> || <math>e</math> || <math>c</math> || <math>b</math>|-|<math>b</math> || <math>b</math> || <math>c</math> || <math>e</math> || <math>a</math>|{#lst:element structure of Klein four-|<math>c</math> || <math>c</math> || <math>b</math> || <math>a</math> || <math>e</math>group|multiplication table}}
{{further|[[element structure of Klein four-group]]}} ===Upto Up to conjugation===
There are four [[conjugacy class]]es, each containing one element (the conjugacy classes are singleton because the group is [[Abelian group|Abelianabelian]].
===Upto Up to automorphism===
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.|p-| [[derived length]] || [[arithmetic function value::derived length;1|1]] || The group is abelian.|-| [[Frattini length]] || [[arithmetic function value::Frattini length;1table|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;underlying prime = 2|2]] || Elementary abelian of rank two.order = 4|-| [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;2|2]] || |order p-| [[max-length of a group|max-length]] || [[arithmetic function value::max-length of a group;log = 2|2]] |||-| [[Rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;2|exponent = 2]] || |-| [[Normal rank of a exponent p-group|normal rank]] || [[arithmetic function value::normal rank of a p-group;2|2]] || log = 1|-| [[characteristic rank of a p-group]] || [[arithmetic function value::characteristic rank of a p-group;2|= 2]] || |}}
==Group properties==
{| class="wikitablesortable" border="1"!Property !! Satisfied ? !! Explanation !! Comment
|[[Satisfies property::Abelian group]] || Yes || ||
|[[Dissatisfies property::Cyclic group]] || No || ||
|[[Satisfies property::Rational-representation group]] || Yes || ||
|[[Satisfies property::Rational group]] || Yes || ||
|[[Satisfies property::Ambivalent group]] || Yes || ||
{{normal subgroupsfurther|[[subgroup structure of Klein four-group]]}}
All subgroups are normal, since the group is Abelian. There is a total of five subgroups: the whole group, the trivial subgroup, and two-element subgroups (viz copies of [[cyclic groupFile:Z2V4latticeofsubgroups.png|the cyclic group of order 2400px]]).
{{characteristic subgroups}} The #lst:subgroup structure of Kleinfour-four group is a [[characteristically simple group]], since it is a direct power of a simple group. Hence, the only characteristic subgroups are the trivial subgroup and the whole group.|summary}}
==Bigger groups==
==Implementation in GAP==
{{GAP ID|4|2}} ===Group IDOther descriptions===
The Klein-four group is the second group of order 4 as per can also be defined using GAP's small-group enumeration, so it can be described in [[GAP :ElementaryAbelianGroup|ElementaryAbelianGroup]] function as:
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