Non-normal Klein four-subgroups 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).
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 Young subgroup for the partition $\{ \{ 1,2 \} , \{ 3,4 \} \}$. Explicitly, it is the subgroup comprising those permutations that send each of the subsets $\{ 1,2 \}$ and $\{ 3,4 \}$ to within itself. $H$ has a total of three conjugates, listed below:

Our local name	Partition stabilized	Set of all elements in the stabilizing subgroup
$\! H$	$\! \{ \{ 1,2 \}, \{ 3,4 \} \}$	$\! \{ (), (1,2), (3,4), (1,2)(3,4) \}$
$\! H_1$	$\! \{ \{ 1,3 \}, \{ 2,4 \} \}$	$\! \{ (), (1,3), (2,4), (1,3)(2,4) \}$
$\! H_2$	$\! \{ \{ 1,4 \}, \{ 2,3 \} \}$	$\! \{ (), (1,4), (2,3), (1,4)(2,3) \}$

$H$ (and hence each of its conjugate subgroups) is isomorphic to the Klein four-group. However, $G$ has another subgroup isomorphic to the Klein four-group that is not one of these conjugate subgroups, and is not automorphic to these either. That is the normal Klein four-subgroup of symmetric group:S4 that comprises the identity element and the three double transpositions. The current article is not about that subgroup.

Cosets

There is a total of 18 cosets, each of which is a left coset for exactly one subgroup and a right coset for exactly one subgroup. Moreover, each coset is parametrized by the way it sends one partition (labeled) to another. Table below is incomplete.

Subset in cycle decomposition notation	Subset in one-line notation	Source and target subsets of $\{ 1,2,3,4 \}$	Left coset of $H$?	Left coset of $H_1$?	Left coset of $H_2$?	Right coset of $H$?	Right coset of $H_1$?	Right coset of $H_2$?
$\{ (), (1,2), (3,4), (1,2)(3,4) \}$	1234, 2134, 1243, 2143	$\{ 1,2 \} \to \{ 1,2 \}$, $\{ 3,4 \} \to \{ 3,4 \}$	Yes	No	No	Yes	No	No
$\{ (), (1,3), (2,4), (1,3)(2,4) \}$	1234, 3214, 1432, 3412	$\{ 1,3 \} \to \{ 1,3 \}$, $\{ 2,4 \} \to \{ 2,4 \}$	No	Yes	No	No	Yes	No
$\{ (), (1,4), (2,3), (1,4)(2,3) \}$	1234, 4231, 1324, 4321	$\{ 1,4 \} \to \{ 1,4 \}$, $\{ 2,3 \} \to \{ 2,3 \}$	No	No	Yes	No	No	Yes
$\{ (1,3,2,4), (1,4,2,3), (1,3)(2,4), (1,4)(2,3) \}$	3421, 4312, 3412, 4321	$\{ 1,2 \} \to \{ 3,4 \}$, $\{ 3,4 \} \to \{ 1, 2 \}$	Yes	No	No	Yes	No	No
$\{ (1,2,3,4), (1,4,3,2), (1,2)(3,4), (1,4)(2,3) \}$	2341, 4123, 2143, 4321	$\{ 1,3 \} \to \{ 2,4 \}$, $\{ 2,4 \} \to \{ 1,3 \}$	No	Yes	No	No	Yes	No
$\{ (1,2,4,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4) \}$	2413, 3142, 2143, 3412	$\{ 1,4 \} \to \{ 2,3 \}$, $\{ 2,3 \} \to \{ 1,4 \}$	No	No	Yes	No	No	Yes

Complements

None of these subgroups has a permutable complement. All the subgroups do have a common lattice complement; in fact, there is a conjugacy class of subgroups each of which is conjugate to each of these subgroups. That conjugacy class is the conjugacy class of A3 in S4, which includes each of these subgroups:

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

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
retract	has a normal complement	No
permutably complemented subgroup	has a permutable complement	No
lattice-complemented subgroup	has a lattice complement	Yes	$\{ (), (1,2,3), (1,3,2) \}$
direct factor	normal subgroup with normal complement	No
complemented normal subgroup	normal subgroup with permutable complement	No

Arithmetic functions

Function	Value	Explanation
order of whole group	24
order of subgroup	4
index of the subgroup	6
size of conjugacy class	3
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	$\{ (), (1,2), (3,4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (1,3,2,4), (1,4,2,3) \}$	D8 in S4	dihedral group:D8
centralizer	the subgroup itself	(current page)	Klein four-group
normal core	trivial subgroup	--	trivial group
normal closure	the whole group	--	symmetric group:S4
characteristic core	trivial subgroup	--	trivial group
characteristic closure	the whole group	--	symmetric group:S4

Subgroup properties

Normality-related properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No	(see above for other conjugate subgroups)
2-subnormal subgroup	normal subgroup in its normal closure	No
subnormal subgroup	series from subgroup to whole group, each normal in next	No
contranormal subgroup	normal closure is whole group	Yes
self-normalizing subgroup	equals normalizer in the whole group	No	normalizer is dihedral group of order eight
self-centralizing subgroup	contains its centralizer in the whole group	Yes
subgroup whose join with any distinct conjugate is the whole group	join of the subgroup with any distinct conjugate subgroup is the whole group	Yes
pronormal subgroup	any conjugate to it is conjugate in their join	Yes
weakly pronormal subgroup		Yes
paranormal subgroup		Yes
polynormal subgroup		Yes

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
order-isomorphic subgroup	isomorphic to every subgroup of the group of the same order	No	The subgroup $\{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}$ has the same order but is isomorphic to cyclic group:Z4
isomorph-automorphic subgroup	any subgroup of the group isomorphic to it is automorphic to it	No	There is also the subgroup $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$	See normal Klein four-subgroup of symmetric group:S4
automorph-conjugate subgroup		Yes