A3 in A4

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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) alternating group:A3 and the group is (up to isomorphism) alternating group:A4 (see subgroup structure of alternating group:A4).
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A4latticeofsubgroups.png

This article describes the subgroup H in the group G. Here, G is the alternating group:A4, acting on the set \{ 1,2,3,4 \}. H is the subgroup:

\! H = H_4 := \{ (), (1,2,3), (1,3,2) \}

It has three other conjugates:

\! H_1 := \{ (), (2,3,4), (2,4,3) \}, H_2 := \{ (), (1,3,4), (1,4,3) \}, H_3 := \{ (), (1,2,4), (1,4,2) \}

With this notation, each H_i is the stabilizer of \{ i \} in G.

See also subgroup structure of alternating group:A4.

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:

\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}

This is also the unique permutable complement to each of them.

Also, each of H_1, H_2, H_3, H_4 has each of the following three subgroups as a lattice complement that is not a permutable complement:

\{ (), (1,2)(3,4) \}, \qquad \{ (), (1,3)(2,4) \}, \qquad \{ (), (1,4)(2,3) \}

Properties related to complementation

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 H.

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 \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} V4 in S4 Klein four-group

Related subgroups

Intermediate subgroups

The subgroup H is a maximal subgroup of G, so there are no strictly intermediate subgroups between H and G.

Smaller subgroups

The subgroup H is a group of prime order, so it has no proper nontrivial subgroup.

Normality-related properties

For ease of reference, we take here the subgroup H = \{ (), (1,2,3), (1,3,2) \}, 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