Difference between revisions of "Klein fourgroup"
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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
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This particular group is the smallest (in terms of order): noncyclic group
This particular group is a finite group of order: 4
Contents
Definition
Verbal definitions
The Kleinfour group is defined in the following equivalent ways:
 It is the direct product of the group with itself
 It is the group comprising the elements under coordinatewise multiplication
 It is the unique noncyclic group of order 4
 It is the subgroup of the symmetric group on 4 elements comprising the double transpositions, and the identity element.
Multiplication table
Element  

Elements
Upto conjugation
There are four conjugacy classes, each containing one element (the conjugacy classes are singleton because the group is Abelian.
Upto automorphism
There are two equivalence classes of elements upto automorphism: the identity element as a singleton, and all the nonidentity elements. All the nonidentity elements are equivalent under automorphism.
Group properties
Abelianness
This particular group is Abelian
Endomorphisms
Automorphisms
The automorphism group is naturally identified with the group as follows. Each element of the automorphism group corresponds to a permutation of the three nonidentity elements.
The holomorph, viz the direct product with the automorphism group, is the symmetric group on 4 elements.
Endomorphisms
The nonautomorphism endomorphisms include:
 The trivial map
 Pick an arbitrary direct sum decomposition and an arbitrary twoelement subgroup. Then the projection on the first direct factor for the decomposition, composed with the isomorphism to the other twoelement 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 twoelement subgroups (viz copies of the cyclic group of order 2).
Characteristic subgroups
The Kleinfour 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.
Bigger groups
Groups containing it as a subgroup
 Alternating group:A4 which is the semidirect product of the Kleinfour group by a cyclic group of order 3
 Symmetric group:S4 which is the holomorph of the Kleinfour group, and in which the Kleinfour 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 Kleinfour group and the symmetric group on 4 elements
Note that the Kleinfour 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 Kleinfour group.
It may also occur as the intersection of indextwo subgroups that are not automorphs of each other.
Some examples:
 The quaternion group, which has the Kleinfour group as its inner automorphism group. The normal subgroups can be taken as those generated by the squareroots of
 The dihedral group of order eight, which has the Kleinfour 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 Kleinfour group.
Implementation in GAP
Group ID
The Kleinfour group is the second group of order 4 as per GAP's smallgroup enumeration, so it can be described in GAP as:
SmallGroup(4,2)