A3 in A4
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) alternating group:A3 and the group is (up to isomorphism) alternating group:A4 (see subgroup structure of alternating group:A4).
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This article describes the subgroup in the group
. Here,
is the alternating group:A4, acting on the set
.
is the subgroup:
It has three other conjugates:
With this notation, each is the stabilizer of
in
.
See also subgroup structure of alternating group:A4.
Contents
Cosets
Each of the four subgroups has four left cosets and four right cosets. Further, for every pair of subgroups, there is exactly one coset that is a left coset for the first subgroup and a right coset for the second subgroup.
Complements
All four subgroups have a unique common normal complement, which is the Klein four-subgroup of alternating group:A4:
This is also the unique permutable complement to each of them.
Also, each of has each of the following three subgroups as a lattice complement that is not a permutable complement:
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
permutably complemented subgroup | has a permutable complement | Yes | See above | |
lattice-complemented subgroup | has a lattice complement | Yes | See above | |
retract | has a normal complement | Yes | See above | |
complemented normal subgroup | normal subgroup with permutable complement | No | Not normal |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 12 | |
order of subgroup | 3 | |
index | 4 | |
size of conjugacy class | 4 | |
number of conjugacy classes in automorphism class | 1 |
Effect of subgroup operators
In the table below, we provide values specific to .
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | the subgroup itself | current page | cyclic group:Z3 |
centralizer | the subgroup itself | current page | cyclic group:Z3 |
normal core | trivial subgroup | -- | trivial group |
normal closure | whole group | -- | alternating group:A4 |
characteristic core | trivial subgroup | -- | trivial group |
characteristic closure | whole group | -- | alternating group:A4 |
commutator with whole group | subgroup ![]() |
V4 in S4 | Klein four-group |
Related subgroups
Intermediate subgroups
The subgroup is a maximal subgroup of
, so there are no strictly intermediate subgroups between
and
.
Smaller subgroups
The subgroup is a group of prime order, so it has no proper nontrivial subgroup.
For ease of reference, we take here the subgroup , though the conclusions apply for the other three conjugates as well.
Some of the properties follow on account of being a Sylow subgroup.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | equals its conjugate subgroups | No | ||
subnormal subgroup | has a chain to whole group each normal in next | No | ||
pronormal subgroup | Yes | |||
automorph-conjugate subgroup | all automorphic subgroups are conjugate subgroups | Yes | ||
isomorph-conjugate subgroup | Yes | |||
order-conjugate subgroup | Yes | |||
order-automorphic subgroup | Yes | |||
order-isomorphic subgroup | Yes |