Klein four-group

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

This particular group is a finite group of order: 4

Definition

Verbal definitions

The Klein four-group, usually denoted $V_4$, 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 of degree four comprising the double transpositions, and the identity element.
• It is the Burnside group $B(2,2)$: the free group on two generators with exponent two.

Multiplication table

Element	$e$	$a$	$b$	$c$
$e$	$e$	$a$	$b$	$c$
$a$	$a$	$e$	$c$	$b$
$b$	$b$	$c$	$e$	$a$
$c$	$c$	$b$	$a$	$e$

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 non-identity elements. All the non-identity elements are equivalent under automorphism.

Arithmetic functions

Function	Value	Explanation
order	4
exponent	2	Cyclic subgroup of order two.
nilpotency class	1	The group is abelian.
derived length	1	The group is abelian.
Frattini length	1	The group is elementary abelian.
Fitting length	1	The group is abelian, hence nilpotent.
minimum size of generating set	2	Elementary abelian of rank two.
subgroup rank	2
max-length	2
rank as p-group	2
normal rank	2
characteristic rank of a p-group	2

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	Yes
Nilpotent group	Yes
Elementary abelian group	Yes
Solvable group	Yes
Supersolvable group	Yes
Cyclic group	No
Rational-representation group	Yes
Rational group	Yes
Ambivalent group	Yes

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.

Bigger groups

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 $-1$
• The 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.

Implementation in GAP

Group ID

The Klein-four group is the second group of order 4 as per GAP's small-group enumeration, so it can be described in GAP as:

SmallGroup(4,2)