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 on
by
. We define:
.
Its other images under inner automorphisms (which are the same as its images under automorphisms) are:
-
.
-
.
Contents
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 | ![]() |
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 |
Related subgroups
Intermediate subgroups
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 |
Smaller subgroups
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 |
Isomorph-conjugate subgroup
Further information: isomorph-conjugate subgroup, isomorph-automorphic subgroup, automorph-conjugate subgroup
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
Opposites satisfied
-
is a core-free subgroup of
: the normal core of
in
is trivial.
-
is a contranormal subgroup of
: the normal closure of
in
is
.
Opposites dissatisfied
-
is not a self-normalizing subgroup of
.
-
is not an abnormal subgroup of
or a weakly abnormal subgroup of
, because both of these would imply that
is a self-normalizing subgroup of
.