Klein four-subgroup of alternating group:A4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) Klein four-group and the group is (up to isomorphism) alternating group:A4 (see subgroup structure of alternating group:A4).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z3.
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This article describes the subgroup $H$ in the group $G$. Here, $G$ is the alternating group of degree four, acting on the set $\{ 1,2,3,4 \}$ for concreteness. In other words, $G$ is the set:

$\! G := \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (1,2,3), (1,3,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3), (1,2,4), (1,4,2) \}$

The subgroup $H$ is defined as the subgroup comprising the identity element and the three double transpositions, which can be characterized as the only even permutations that have no fixed points:

$\! H := \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$

The subgroup is isomorphic to the Klein four-group. It is a normal subgroup and the quotient group is isomorphic to cyclic group:Z3.

See also subgroup structure of alternating group:A4.

Cosets

$H$ is a normal subgroup of $G$, hence each left coset is a right coset and vice versa. The three cosets are then:

$\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}, \qquad \{ (1,2,3), (2,4,3), (1,4,2), (1,3,4) \}, \qquad \{ (1,3,2), (2,3,4), (1,2,4), (1,4,3) \}$

Some interesting observations:

• Each of the cosets other than the subgroup itself is itself a conjugacy class. This makes the alternating group a Camina group.
• Each 3-cycle and its inverse are contained in distinct cosets of $H$. This makes sense, if we note that the cyclic subgroup generated by the 3-cycle must be a permutable complement to $H$ in $G$.

Complements

COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)

$H$ is a complemented normal subgroup in $G$. There are four distinct possibilities for the complement of $H$ in $G$, all of which are conjugate subgroups. This also follows from the Schur-Zassenhaus theorem, since $H$ is a normal Sylow subgroup of $G$.

The four complements are:

$\{ (), (1,2,3), (1,3,2) \}, \qquad \{ (), (1,3,4), (1,4,3) \}, \qquad \{ (), (2,4,3), (2,3,4) \}, \qquad \{ (), (1,4,2), (1,2,4) \}$

For information on these as subgroups, see A3 in A4. Each of these is isomorphic to cyclic group:Z3, since complement to normal subgroup is isomorphic to quotient.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
complemented normal subgroup	normal subgroup with a permutable complement	Yes	See above
complemented characteristic subgroup	characteristic subgroup with a permutable complement	Yes
complemented fully invariant subgroup	fully invariant subgroup with a permutable complement	Yes
permutably complemented subgroup	subgroup with a permutable complement	Yes
lattice-complemented subgroup	subgroup with lattice complement	Yes
retract	subgroup with a normal complement	No
direct factor	normal subgroup with a normal complement	No

Arithmetic functions

Function	Value	Explanation
order of whole group	12
order of subgroup	4
index	3
size of conjugacy class	1
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the whole group	--	alternating group:A4
centralizer	the subgroup itself	current page	Klein four-group
normal core	the subgroup itself	current page	Klein four-group
normal closure	the subgroup itself	current page	Klein four-group
characteristic core	the subgroup itself	current page	Klein four-group
characteristic closure	the subgroup itself	current page	Klein four-group
commutator with whole group	the subgroup itself	current page	Klein four-group

Subgroup-defining functions

The subgroup is a characteristic subgroup of the whole group and arises as the result of many subgroup-defining functions. Some of these are given below.

Subgroup-defining function	Meaning in general	Why it takes this value
derived subgroup	subgroup generated by commutators of all pairs of group elements, smallest subgroup with abelian quotient	The quotient is cyclic group:Z3, which is abelian; no other subgroup has abelian quotient. We can also explicitly compute all commutators -- these are precisely the identity element and the three double transpositions.
socle	join of all minimal normal subgroups	The subgroup is the unique minimal normal subgroup (i.e., monolith) -- the group is a monolithic group
Fitting subgroup	join of all nilpotent normal subgroups	The subgroup is the unique nontrivial abelian normal subgroup
2-Sylow core	largest normal subgroup whose order is a power of 2; normal core of any 2-Sylow subgroup	The subgroup is the unique normal 2-Sylow subgroup
2-Sylow closure	normal closure of any 2-Sylow subgroup	The subgroup is the unique normal 2-Sylow subgroup
Jacobson radical	intersection of all maximal normal subgroups	The subgroup is the unique maximal normal subgroup -- the group is a one-headed group

Description in alternative interpretations of the whole group

Description of $G$	Corresponding description of $H$
Group of orientation-preserving symmetries of a regular tetrahedron	The identity element and the three symmetries that arise as follows: take two opposite sides of the regular tetrahedron (i.e., edges not sharing a common vertex). Consider the line joining the midpoints of these sides. Now, take the half-turn (i.e. rotation by an angle of $\pi$) about this line as axis.
Projective special linear group of degree two over field:F3	The images of the semisimple elements. Note that in general, the images of the semisimple elements do not form a subgroup, this is a special feature of the small prime.

Related subgroups

Intermediate subgroups

The subgroup has prime index, hence is maximal, so there are no strictly intermediate subgroups between the subgroup and the whole group.

Smaller subgroups

There are three proper nontrivial subgroups of the subgroup, all of which are conjugate inside the whole group (but not within the subgroup, which is abelian). The three subgroups are:

$\{ (), (1,2)(3,4) \}, \qquad \{ (), (1,3)(2,4) \}, \qquad \{ (), (1,4)(2,3) \}$

Each of these is isomorphic to cyclic group:Z2. For information on these as subgroups inside $H$, see Z2 in V4. For information on these as subgroups inside $G$, see subgroup generated by double transposition in A4.

Images under quotient maps

Under any quotient map with a nontrivial kernel, the image of the subgroup is trivial. This is because the group is a monolithic group and the subgroup is the unique minimal normal subgroup in it.

Invariance under automorphisms and endomorphisms: properties

Unless otherwise specified, each of these properties follows on account of the subgroup being a normal Sylow subgroup and in particular a normal Hall subgroup.

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	Yes
characteristic subgroup	invariant under all automorphisms	Yes
coprime automorphism-invariant subgroup	invariant under all coprime automorphisms, i.e., automorphisms whose order is coprime to that of the group	Yes
cofactorial automorphism-invariant subgroup	invariant under all cofactorial automorphisms, i.e., automorphisms whose order has no prime factors other than those in the group	Yes
subgroup-cofactorial automorphism-invariant subgroup	invariant under all automorphisms whose order has no prime factors other than those in the subgroup	Yes
order-conjugate subgroup	conjugate to all subgroups of the same order	Yes	follows from being a Sylow subgroup, since Sylow implies order-conjugate
order-isomorphic subgroup	isomorphic to all subgroups of the same order	Yes	(via order-conjugate, also obvious since has prime order)
isomorph-free subgroup	no other isomorphic subgroup	Yes
order-unique subgroup	no other subgroup of that order	Yes
fully invariant subgroup	contains its image under any endomorphism of whole group	Yes	normal Sylow subgroups are fully invariant
verbal subgroup		Yes
normal subgroup having no nontrivial homomorphism to its quotient group		Yes
homomorph-containing subgroup	contains any homomorphic image of it in the whole group	Yes	normal Sylow subgroups are homomorph-containing
variety-containing subgroup		Yes
quotient-subisomorph-containing subgroup		Yes