Supergroups of Klein four-group
This article gives specific information, namely, supergroups, about a particular group, namely: Klein four-group.
View supergroups of particular groups | View other specific information about Klein four-group
This article discusses possible supergroups of the Klein four-group.
Note that unlike the discussion of the subgroup structure of Klein four-group, thisdiscussion is necessarily not comprehensive, because there are infinitely many groups containing the Klein four-group as a subgroup.
Each of the groups listed below arises as the external direct product of Klein four-group and some nontrivial group. In particular, each of these groups contains the Klein four-group as a direct factor -- and hence as both a normal subgroup and a quotient group.
Note that since order of direct product is product of orders, if the other group has order , the direct product has order .
|Other factor of direct product||Order||Second part of GAP ID||Value of direct product||Order||Second part of GAP ID||Hall-Senior symbol (if applicable)|
|cyclic group:Z2||2||1||elementary abelian group:E8||8||5|
|cyclic group:Z3||3||1||direct product of Z6 and Z2||12||5|
|cyclic group:Z4||4||1||direct product of Z4 and V4||16||10|
|Klein four-group||4||2||elementary abelian group:E16||16||14|
|cyclic group:Z5||5||1||direct product of Z10 and Z2||20||5|
|symmetric group:S3||6||1||direct product of D12 and Z2||24||14|
|cyclic group:Z6||6||2||direct product of E8 and Z3||24||15|
Classification of supergroups of order eight
The general procedure
Suppose is a group of order eight containing a subgroup of order four isomorphic to the Klein four-group. Note that since is of index two, it is a normal subgroup. Further, the quotient group is isomorphic to , the cyclic group of order two.
The classification has two steps:
- Find all homomorphisms from to .
- For each such homomorphism, find all possible extensions. These are classified by the elements of for the action.
is isomorphic to the symmetric group of degree three. This has three subgroups of order two, all of which are conjugate. Thus, up to equivalence under automorphisms of , there is only one subgroup of order two.
Thus, there are (up to equivalence under automorphisms) two homomorphisms from to : the trivial homomorphism and an isomorphism to one of the subgroups of order two.
For the trivial map, the cohomology group has order two, and the two extensions are:
- elementary abelian group of order eight: This corresponds to the identity element of .
- direct product of Z4 and Z2: This corresponds to the non-identity element of .
For the nontrivial map, the cohomology group is trivial, and the unique extension corresponding to it is the dihedral group of order eight.