# Changes

## Klein four-group

, 16:32, 21 December 2014
Subgroups
{{particular group}}
{{smallest|non-[[cyclic groupImportance rank::1| ]]}} {{group of order|4}}
==Definition==
===Verbal definitions===
The Klein four-group , usually denoted $V_4$, is defined in the following equivalent ways:
* It is the [[direct product]] of the group $\mathbb{Z}/2\mathbb{Z}$ with itself
* It is the group comprising the elements $(\pm 1, \pm 1)$ 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 on 4 elementsof degree four]] comprising the double transpositions, and the identity element.* It is the [[member of family::Burnside group]] $B(2,2)$: the ''free group'' on two generators with exponent two.
===Multiplication table===
{| class="wikitable" border="1"! Element !! $e$ !! $a$ !! $b$ !! $c$|-|$e$ || $e$ || $a$ || $b$ || $c$|-|$a$ || $a$ || $e$ || $c$ || $b$|-|$b$ || $b$ || $c$ || $e$ || $a$|{#lst:element structure of Klein four-|$c$ || $c$ || $b$ || $a$ || $e$group|multiplication table}}
==Elements==
{{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==

{{abelian p-group arithmetic function table|
underlying prime = 2|
order = 4|
order p-log = 2|
exponent = 2|
exponent p-log = 1|
rank = 2}}
==Group properties==
{{not cyclic}}| class="sortable" 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 || |||-|[[Satisfies property::Rational-representation group]] || Yes || |||-|[[Satisfies property::Rational group]] || Yes || |||-|[[Satisfies property::Ambivalent group]] || Yes || |||}
==Endomorphisms==
==Subgroups==
{{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:
<prett>SmallGroupElementaryAbelianGroup(4,2)</prett>