# Changes

## Klein four-group

, 00:14, 24 September 2007
Groups containing it as a subgroup
The Klein-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.
==Groups containing it as a subgroupBigger groups==
These include:===Groups containing it as a subgroup===
* [[Alternating group:A4]] which is the semidirect product of the Klein-four group by a cyclic group of order 3
* [[Symmetric group:S4]] which is the holomorph of the Klein-four group, and in which the Klein-four group is a [[characteristic subgroup]]
* [[Dihedral group:D8]] which is the dihedral group of order 8, acting on a set of four elements. It sits between the Klein-four group and the symmetric group on 4 elements

Note that the Klein-four group embeds in two ways inside the symmetric group, one, as double transpositions, the other, as the direct product of a pair of involutions. We usually refer to the former embedding, when nothing is explicitly stated.

===Groups having it as a quotient===

In general, whenever a group has a subgroup of index two that is ''not'' characteristic, then the intersection of that subgroup and any other automorph of it, is of index four, and the quotient obtained is the Klein-four group.

It may also occur as the intersection of index-two subgroups that are not automorphs of each other.

Some examples:

* The [[quaternion group]], which has the Klein-four group as its [[inner automorphism group]]. The normal subgroups can be taken as those generated by the squareroots of [itex]-1[/itex]
* The [[dihedral group:D8|dihedral group of order eight]], which has the Klein-four group as its [[inner automorphism group]]. Here, it is the quotient by the intersection of two subgroups of order four, one being a cyclic subgroup, the other being itself a Klein-four group.