Cyclic four-subgroups of symmetric group:S4

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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) 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:

Opposites of normality satisfied and dissatisfied

Opposites satisfied

Opposites dissatisfied