# Changes

## Klein four-group

, 16:32, 21 December 2014
Subgroups
{{particular group}}
[[Importance rank::1| ]]
==Definition==
===Multiplication table===
{| class="sortable" border="1"! Element !! [itex]e[/itex] !! [itex]a[/itex] !! [itex]b[/itex] !! [itex]c[/itex]|-|[itex]e[/itex] || [itex]e[/itex] || [itex]a[/itex] || [itex]b[/itex] || [itex]c[/itex]|-|[itex]a[/itex] || [itex]a[/itex] || [itex]e[/itex] || [itex]c[/itex] || [itex]b[/itex]|-|[itex]b[/itex] || [itex]b[/itex] || [itex]c[/itex] || [itex]e[/itex] || [itex]a[/itex]|{#lst:element structure of Klein four-|[itex]c[/itex] || [itex]c[/itex] || [itex]b[/itex] || [itex]a[/itex] || [itex]e[/itex]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.
{{further|[[subgroup structure of Klein four-group]]}}
{{normal subgroups}} 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==