# Klein four-subgroup of alternating group:A4

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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.

## 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