Normal Klein four-subgroup of symmetric group:S4

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) symmetric group:S4 (see subgroup structure of symmetric group:S4).
The subgroup is a normal subgroup and the quotient group is isomorphic to symmetric group:S3.
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This article discusses the normal subgroup in the symmetric group of degree four comrpising the identity and the three double transpositions.

We let $G = S_4$ be the symmetric group of degree four, acting on $\{ 1,2,3,4 \}$ and $H$ be the subgroup of $G$ given by:

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

Cosets

The six cosets of this subgroup are as follows:

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

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

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)

There are four possible complements to $H$ in $G$, all of which are conjugate subgroups and all are isomorphic to symmetric group:S3. These are the stabilizers of the individual points $1,2,3,4$:

Fixed point	Subgroup
1	$\! \{ (), (2,3,4), (2,4,3), (2,3), (3,4), (2,4) \}$
2	$\! \{ (), (1,3,4), (1,4,3), (1,3), (3,4), (1,4) \}$
3	$\! \{ (), (1,2,4), (1,4,2), (1,2), (2,4), (1,4) \}$
4	$\! \{ (), (1,2,3), (1,3,2), (1,2), (2,3), (1,3) \}$

Each of these is isomorphic to the quotient group $G/H$, since complement to normal subgroup is isomorphic to quotient.

Properties related to complementation

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

Arithmetic functions

Function	Value	Explanation
order of whole group	24
order of subgroup	4
index	6
size of conjugacy class	1
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

In the table below, we provide values specific to $H$.

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	whole group	--	symmetric group:S4
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

Subgroup-defining functions

The subgroup is a characteristic subgroup of the whole group and arises as the result of many common subgroup-defining functions on the whole group.

Subgroup-defining function	What it means in general	Why it takes this value
second derived subgroup	second member of derived series. In other words, derived subgroup of derived subgroup, i.e., $[[G,G],[G,G]]$	The derived subgroup is A4 in S4, and the derived subgroup of the derived subgroup is the Klein four-subgroup. The corresponding quotient groups are cyclic group:Z2 and cyclic group:Z3.
2-Sylow core or 2-core	largest normal subgroup whose order is a power of 2. Equivalently, normal core of any 2-Sylow subgroup	There are three 2-Sylow subgroups, each isomorphic to dihedral group:D8, and their intersection is this subgroup.
socle	join of all minimal normal subgroups	In fact, it is the unique minimal normal subgroup (hence the monolith) -- the whole group is a monolithic group.

Description in terms of alternative interpretations of the whole group

Interpretation of $G$	Corresponding interpretation of $H$
As symmetric group of degree four	Subset comprising identity and double transpositions
As projective general linear group of degree two over field:F3	images of semisimple elements of determinant 1 (note: this does not usually form a subgroup, but does for this field and this degree)
As group of all isometries of the regular tetrahedron (including orientation-reversing ones)	identity, and isometries obtained as follows: consider two opposite edges (i.e., edges that share no vertex). Take the line joining their midpoints. Now, do a half turn (rotation by 180 degrees) about this line.
As group of all orientation-preserving isometries of the cube, or the regular octahedron	identity, and rotations by 180 degrees about the three axes of the cube.

Related subgroups

Intermediate subgroups

Value of intermediate subgroup (descriptive)	Isomorphism class of intermediate subgroup	Number of conjugacy classes of intermediate subgroup fixing subgroup and whole group	Subgroup in intermediate subgroup	Intermediate subgroup in whole group
$\langle (1,2)(3,4), (1,2,3,4) \rangle$	dihedral group:D8	3	Klein four-subgroups of dihedral group:D8	D8 in S4
$\langle (1,2)(3,4), (1,2,3) \rangle$	alternating group:A4	1	Klein four-subgroup of alternating group:A4	A4 in S4

Smaller subgroups

Value of smaller subgroup (descriptive)	Isomorphism class of smaller subgroup	Number of conjugacy classes of smaller subgroup fixing subgroup and whole group	Smaller subgroup in subgroup	Smaller subgroup in whole group
$\{ (), (1,2)(3,4) \}$, $\{ (), (1,3)(2,4) \}$, $\{ (), (1,4)(2,3) \}$	cyclic group:Z2	3	Z2 in V4	subgroup generated by double transposition in S4

Invariance under automorphisms and endomorphisms: properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	invariant under all inner automorphisms, equals all conjugate subgroups	Yes	identity element plus conjugacy class of all double transpositions
characteristic subgroup	invariant under all automorphisms	Yes	Since the whole group is a complete group, all automorphisms are inner
fully invariant subgroup	invariant under all endomorphisms	Yes	Since it is the second derived subgroup; also because monolith is fully invariant in co-Hopfian
verbal subgroup	set of all occurrences of a bunch of words	Yes	Since it is the second derived subgroup
image-closed fully invariant subgroup	under any homomorphic image of whole group, image is fully invariant	Yes	verbal implies image-closed fully invariant
image-closed characteristic subgroup	under any homomorphic image of whole group, image is characteristic	Yes	(Via image-closed fully invariant)
homomorph-containing subgroup	contains any homomorphic image of it in the whole group	No	There are cyclic subgroups of order two in the whole group not in the subgroup, namely, those generated by single transpositions. These cyclic subgroups are homomorphic images of $H$.
isomorph-containing subgroup, isomorph-free subgroup	contains/equals every isomorphic subgroup (properties equivalent for finite groups)	No	The subgroup $\{ (), (1,2), (3,4), (1,2)(3,4) \}$ and its conjugates, non-normal Klein four-subgroups of symmetric group:S4, are isomorphic to it.
intermediately characteristic subgroup	characteristic in every intermediate subgroup	No	Not characteristic in $\langle (1,2)(3,4), (1,2,3,4) \rangle$, where it sits in like Klein four-subgroups of dihedral group:D8

Interpretations from subgroup upwards

Interpretation in terms of Cayley's theorem

We can think of the embedding of $H$ in $G$ in terms of Cayley's theorem. Specifically, think of starting with $H$ as an abstract Klein four-group whose elements are labeled $1,2,3,4$. Left multiplication by elements of $H$ induces precisely the identity and the three double transpositions on $H$ as a set. This thus makes $H$ a subgroup of the symmetric group on $\{ 1,2,3,4 \}$, which is $G$.

Note that the other, non-normal Klein four-subgroup cannot be interpreted this way because its non-identity elements are not fixed-point-free. However, the cyclic four-subgroup of the symmetric group of degree four can be embedded in this way.

Interpretation in terms of holomorph

We can think of the embedding of $H$ in $G$ as the abstract group $H$ sitting inside its holomorph. This is because $G$ is the semidirect product of $H$ and its automorphism group, which is $GL(2,2)$, which is isomorphic to the symmetric group of degree three.