Cyclic 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) cyclic group:Z4 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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This article is about a conjugacy class (also an equivalence class up to automorphisms) in the symmetric group of degree four, comprising cyclic groups of order four.

We denote the symmetric group $S_4$ on $\{ 1,2,3,4 \}$ by $G$. We define:

$H := \langle (1,2,3,4) \rangle = \langle (1,4,3,2) \rangle = \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}$.

Its other images under inner automorphisms (which are the same as its images under automorphisms) are:

• $\langle (1,3,2,4) \rangle = \langle (1,4,2,3) \rangle = \{ (), (1,3,2,4), (1,2)(3,4), (1,4,2,3) \}$.
• $\langle (1,3,4,2) \rangle = \langle (1,2,4,3) \rangle = \{ (), (1,3,4,2), (1,4)(2,3), (1,2,4,3) \}$.

Arithmetic functions

Function	Value	Explanation
order of whole group	24
order of subgroup	4
index	6
size of conjugacy class	3

Effect of subgroup operators

Function	Value as subgroup (descriptive)	value as subgroup (link)	Value as group
normalizer	$\langle (1,2,3,4), (1,3) \rangle$	D8 in S4	dihedral group:D8
centralizer	$\langle (1,2,3,4) \rangle$	same as $H$	cyclic group:Z4
normal core	$()$	trivial subgroup	trivial group
normal closure	$G$	whole group	symmetric group:S4

Related subgroups

Intermediate subgroups

Value of intermediate subgroup (descriptive)	Isomorphism class of intermediate subgroup	Small subgroup in intermediate subgroup	Intermediate subgroup in big group
$\langle (1,2,3,4), (1,3) \rangle$	dihedral group:D8	cyclic maximal subgroup of dihedral group:D8	D8 in S4

Smaller subgroups

Value of smaller subgroup (descriptive)	Isomorphism class of smaller subgroup	Smaller subgroup in subgroup	Smaller subgroup in whole group
$\langle (1,3)(2,4) \rangle$	cyclic group:Z2	Z2 in Z4

Resemblance-related properties satisfied

Isomorph-conjugate subgroup

Further information: isomorph-conjugate subgroup, isomorph-automorphic subgroup, automorph-conjugate subgroup

$H$ is an isomorph-conjugate subgroup of $G$: it is conjugate to all the other subgroups isomorphic to it. Hence, it is also an isomorph-automorphic subgroup and an automorph-conjugate subgroup. Note that since $G$ is a complete group (symmetric groups are complete) all subgroups of $G$ are automorph-conjugate.

Subgroup whose join with any distinct conjugate is the whole group

Further information: subgroup whose join with any distinct conjugate is the whole group

The join of $H$ and any conjugate of $H$ distinct from $H$ is the whole group. In particular, this forces that:

• $H$ is a pronormal subgroup and hence also a weakly pronormal subgroup.
• Since $H$ is an automorph-conjugate subgroup, this also forces that $H$ is a procharacteristic subgroup and weakly procharacteristic subgroup.

Opposites of normality satisfied and dissatisfied

Opposites satisfied

• $H$ is a core-free subgroup of $G$: the normal core of $H$ in $G$ is trivial.
• $H$ is a contranormal subgroup of $G$: the normal closure of $H$ in $G$ is $G$.

Opposites dissatisfied

• $H$ is not a self-normalizing subgroup of $G$.
• $H$ is not an abnormal subgroup of $G$ or a weakly abnormal subgroup of $G$, because both of these would imply that $H$ is a self-normalizing subgroup of $G$.