S2 in 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) cyclic group:Z2 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

We consider the subgroup $H$ in the group $G$ defined as follows.

$G$ is the symmetric group of degree four, which, for concreteness, we take as the symmetric group on the set $\{1,2,3,4\}$ .

$H$ is the subgroup of $G$ comprising those permutations that fix $\{3,4\}$ pointwise. In particular, $H$ is the symmetric group on $\{1,2\}$ , embedded naturally in $G$ . It is isomorphic to cyclic group:Z2. As a set, $H$ contains two elements: $()$ and $(1,2)$ .

There are five other conjugate subgroups to $H$ in $G$ (so the total conjugacy class size of subgroups is 3). Each subgroup fixed pointwise a subset of size two. Equivalently, each subgroup comprises the identity element and a 2-transposition. Specifically, $H$ and its two other conjugate subgroups are:

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

Cosets

Each of these subgroups has 12 left cosets and 12 right cosets. Further, every left coset of one subgroup is a right coset of one of its conjugate subgroups. Overall, there are thus 72 cosets. Each coset is characterized by a fixed behavior on two of the four points.

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Complements

There is a unique permutable complement to all these subgroups, which is also a normal subgroup and hence a normal complement. In particular, each of the subgroups is a retract. The unique permutable complement is A4 in S4, i.e., the alternating group of degree four.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation
retract	has a normal complement	Yes	alternating group:A4 (see A4 in S4) is a complement.
permutably complemented subgroup	has a permutable complement	Yes	(via retract)
lattice-complemented subgroup	has a lattice complement	Yes	(via retract)
direct factor	normal subgroup with normal complement	No	not normal
complemented normal subgroup	normal subgroup with permutable complement	No	not normal

Arithmetic functions

Function	Value	Explanation
order of whole group	24
order of subgroup	2
index of the subgroup	12
size of conjugacy class	6
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	$\{(),(1,2),(3,4),(1,2)(3,4)\}$	non-normal Klein four-subgroups of symmetric group:S4	Klein four-group
centralizer	$\{(),(1,2),(3,4),(1,2)(3,4)\}$	non-normal Klein four-subgroups of symmetric group:S4	Klein four-group
normal core	trivial subgroup	trivial	trivial group
normal closure	whole group	--	symmetric group:S4
characteristic core	trivial subgroup	trivial	trivial group
characteristic closure	whole group	--	symmetric group:S4

Related subgroups

Intermediate subgroups

The values given here are specific to $H$ .

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
$\{(),(1,2),(3,4),(1,2)(3,4)\}$	Klein four-group	1	Z2 in V4	non-normal Klein four-subgroups of symmetric group:S4
$\{(),(1,2),(2,3),(1,3),(1,2,3),(1,3,2)\}$	symmetric group:S3	3	S2 in S3	S3 in S4
$\{(),(1,2),(3,4),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,3,2,4),(1,4,2,3)\}$	dihedral group:D8	1	non-normal subgroups of dihedral group:D8	D8 in S4

Smaller subgroups

The subgroup has order two, hence is minimal, and has no smaller nontrivial subgroups.

Subgroup properties

Normality-related properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No	See list of conjugate subgroups
2-subnormal subgroup	normal in its normal closure	No	Normal closure is whole group
subnormal subgroup	there is a series from the subgroup to the whole group, each normal in the next	No
contranormal subgroup	normal closure is the whole group	Yes	Normal closure is whole group	transpositions generate the finitary symmetric group
self-normalizing subgroup	equals its normalizer in the whole group	No	Normalizer is $\{(),(1,2),(3,4),(1,2)(3,4)\}$
hypernormalized subgroup	taking normalizer repeatedly reaches the whole group	No	Normalizer is self-normalizing
pronormal subgroup	any conjugate subgroup to the subgroup is conjugate to it in their join	No	The subgroup $\{(),(3,4)\}$ is a conjugate, but in their join $\{(),(1,2),(3,4),(1,2)(3,4)\}$ , which is abelian, the two subgroups are not conjugates.
weakly pronormal subgroup		No
paranormal subgroup		No
polynormal subgroup		No
weakly normal subgroup		No

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
order-isomorphic subgroup	any subgroup of the whole group of the same order is isomorphic to it	Yes	any subgroup of order two is isomorphic to cyclic group:Z2
isomorph-automorphic subgroup	any subgroup of the whole group isomorphic to it is related to it by an automorphism	No	The subgroup $\{(),(1,2)(3,4)\}$ is isomorphic but there is no automorphism sending $H$ to this subgroup.	This other subgroup is subgroup generated by double transposition in symmetric group:S4.
automorph-conjugate subgroup	any subgroup automorphic to it is conjugate to it	Yes	The whole group is a complete group -- all automorphisms of it are inner automorphisms.	See symmetric groups are complete