Difference between revisions of "Klein four-group"

From Groupprops
Jump to: navigation, search
 
Line 42: Line 42:
  
 
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.
 
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 subgroup==
 +
 +
These include:
 +
 +
* [[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

Revision as of 15:11, 16 June 2007

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

This particular group is the smallest (in terms of order): non-cyclic group

Definition

The Klein-four group 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 the symmetric group on 4 elements comprising the double transpositions, and the identity element.

Group properties

Template:Not cyclic

Abelianness

This particular group is Abelian

Endomorphisms

Automorphisms

The automorphism group is naturally identified with the group S_3 as follows. Each element of the automorphism group corresponds to a permutation of the three non-identity elements.

The holomorph, viz the direct product with the automorphism group, is the symmetric group on 4 elements.

Endomorphisms

The non-automorphism endomorphisms include:

  • The trivial map
  • Pick an arbitrary direct sum decomposition and an arbitrary two-element subgroup. Then the projection on the first direct factor for the decomposition, composed with the isomorphism to the other two-element subgroup, is an endomorphism.

Subgroups

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 the cyclic group of order 2).

Characteristic subgroups

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 subgroup

These include:

  • 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