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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We denote the symmetric group on by . We define:
Its other images under inner automorphisms (which are the same as its images under automorphisms) are:
- 1 Arithmetic functions
- 2 Effect of subgroup operators
- 3 Related subgroups
- 4 Resemblance-related properties satisfied
- 5 Opposites of normality satisfied and dissatisfied
|order of whole group||24|
|order of subgroup||4|
|size of conjugacy class||3|
Effect of subgroup operators
|Function||Value as subgroup (descriptive)||value as subgroup (link)||Value as group|
|normalizer||D8 in S4||dihedral group:D8|
|centralizer||same as||cyclic group:Z4|
|normal core||trivial subgroup||trivial group|
|normal closure||whole group||symmetric group:S4|
|Value of intermediate subgroup (descriptive)||Isomorphism class of intermediate subgroup||Small subgroup in intermediate subgroup||Intermediate subgroup in big group|
|dihedral group:D8||cyclic maximal subgroup of dihedral group:D8||D8 in S4|
|Value of smaller subgroup (descriptive)||Isomorphism class of smaller subgroup||Smaller subgroup in subgroup||Smaller subgroup in whole group|
|cyclic group:Z2||Z2 in Z4|
is an isomorph-conjugate subgroup of : 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 is a complete group (symmetric groups are complete) all subgroups of 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 and any conjugate of distinct from is the whole group. In particular, this forces that:
- is a pronormal subgroup and hence also a weakly pronormal subgroup.
- Since is an automorph-conjugate subgroup, this also forces that is a procharacteristic subgroup and weakly procharacteristic subgroup.
Opposites of normality satisfied and dissatisfied
- is a core-free subgroup of : the normal core of in is trivial.
- is a contranormal subgroup of : the normal closure of in is .