Klein four-group

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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

The multiplication table with non-identity elements $a,b,c$ and identity element $e$:

Element/element	$e$	$a$	$b$	$c$
$e$	$e$	$a$	$b$	$c$
$a$	$a$	$e$	$c$	$b$
$b$	$b$	$c$	$e$	$a$
$c$	$c$	$b$	$a$	$e$

The multiplication table can be described as follows (and this characterizes the group):

• The product of the identity element and any element is that element itself.
• The product of any non-identity element with itself is the identity element.
• The product of two distinct non-identity elements is the third non-identity element.

Elements

Further information: element structure of Klein four-group

Up to conjugation

There are four conjugacy classes, each containing one element (the conjugacy classes are singleton because the group is abelian.

Up to 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

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 4#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	4	groups with same order
prime-base logarithm of order	2	groups with same prime-base logarithm of order
max-length of a group	2		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	2		chief length equals prime-base logarithm of order for group of prime power order
composition length	2		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	2	groups with same order and exponent of a group | groups with same prime-base logarithm of order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	1	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length	1	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length	Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group	same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group	2	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group	2	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group	2	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class	1		The group is a nontrivial abelian group
derived length	1		The group is a nontrivial abelian group
Fitting length	1		The group is a nontrivial abelian group -

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

Further information: subgroup structure of Klein four-group

Quick summary

Item	Value
Number of subgroups	5 As elementary abelian group of prime-square order for prime $p = 2$: $p + 3 = 2 + 3 = 5$
Number of conjugacy classes of subgroups	5 (same as number of subgroups, because the group is an abelian group
Number of automorphism classes of subgroups	3 As elementary abelian group of order $p^n, p = 2, n = 2$: $n + 1 = 2 + 1 = 3$
Isomorphism classes of subgroups	trivia group (1 time), cyclic group:Z2 (3 times, all in the same automorphism class), Klein four-group (1 time).

Table classifying subgroups up to automorphism

Note that because abelian implies every subgroup is normal, all the subgroups are normal subgroups.

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes(=1 iff automorph-conjugate subgroup)	Size of each conjugacy class(=1 iff normal subgroup)	Total number of subgroups(=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth	Nilpotency class
trivial subgroup	$\{ e \}$	trivial group	1	4	1	1	1	Klein four-group	1	0
Z2 in V4	$\{ e, a\}, \{ e,b \}, \{ e,c \}$	cyclic group:Z2	2	2	3	1	3	cyclic group:Z2	1	1
whole group	$\{ e,a,b,c\}$	Klein four-group	4	1	1	1	1	trivial group	0	1
Total (3 rows)	--	--	--	--	5	--	5	--	--	--

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

This finite group has order 4 and has ID 2 among the groups of order 4 in GAP's SmallGroup library. For context, there are groups of order 4. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(4,2)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(4,2);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [4,2]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

The group can also be defined using GAP's ElementaryAbelianGroup function as:

ElementaryAbelianGroup(4)