Subgroup structure of alternating group:A4: Difference between revisions
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! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! [[Order of a group|Order]] of subgroups !! [[Index of a subgroup|Index]] of subgroups !! Number of conjugacy classes !! Size of each conjugacy class !! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if subnormal) | ! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! [[Order of a group|Order]] of subgroups !! [[Index of a subgroup|Index]] of subgroups !! Number of conjugacy classes !! Size of each conjugacy class !! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if subnormal) | ||
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| trivial subgroup || <math>\{ () \}</math> [[trivial group]] || 1 || 1 || [[alternating group:A4]] || 1 | | trivial subgroup || <math>\{ () \}</math> || [[trivial group]] || 1 || 12 || 1 || 1 || [[alternating group:A4]] || 1 | ||
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| [[subgroup generated by double transposition in A4]] || <math>\{ (), (1,2)(3,4) \}</math>, <math>\{ (), (1,3)(2,4) \}</math>, <math>\{ (), (1,4)(2,3) \}</math> || [[cyclic group:Z2]] || 2 || 6 || 1 || 3 || -- || 2 | | [[subgroup generated by double transposition in A4]] || <math>\{ (), (1,2)(3,4) \}</math>, <math>\{ (), (1,3)(2,4) \}</math>, <math>\{ (), (1,4)(2,3) \}</math> || [[cyclic group:Z2]] || 2 || 6 || 1 || 3 || -- || 2 | ||
Revision as of 16:48, 22 June 2011
This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A4.
View subgroup structure of particular groups | View other specific information about alternating group:A4
The alternating group on is a group of order 12.
There is no subgroup of order . The alternating group of degree four is the group of smallest possible order (in this case ) not having subgroups of all orders dividing the group order.
Tables for quick information
Table classifying subgroups up to automorphism
| Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) |
|---|---|---|---|---|---|---|---|---|
| trivial subgroup | trivial group | 1 | 12 | 1 | 1 | alternating group:A4 | 1 | |
| subgroup generated by double transposition in A4 | , , | cyclic group:Z2 | 2 | 6 | 1 | 3 | -- | 2 |
| V4 in A4 | Klein four-group | 4 | 3 | 1 | 1 | cyclic group:Z3 | 1 | |
| A3 in A4 | , , , <math>\{ (), (1,2,3), (1,3,2) \}</cyclic group:Z3 | 3 | 4 | 1 | 4 | -- | -- | |
| whole group | all elements | alternating group:A4 | 12 | 1 | 1 | 1 | trivial group | 1 |
Table classifying isomorphism types of subgroups
| Group name | Order | Second part of GAP ID (first part is order) | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
|---|---|---|---|---|---|---|---|
| trivial group | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| cyclic group:Z2 | 2 | 1 | 3 | 1 | 1 | 0 | 0 |
| cyclic group:Z3 | 3 | 1 | 4 | 1 | 0 | 0 | |
| Klein four-group | 4 | 2 | 1 | 1 | 1 | 1 | 1 |
| alternating group:A4 | 12 | 3 | 1 | 1 | 1 | 1 | 1 |
| Total | -- | -- | 10 | 5 | 5 | 3 | 3 |
Table listing numbers of subgroups by group property
| Group property | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
|---|---|---|---|---|---|
| cyclic group | 8 | 3 | 3 | 1 | 1 |
| abelian group | 9 | 4 | 4 | 2 | 2 |
| nilpotent group | 9 | 4 | 4 | 2 | 2 |
| solvable group | 10 | 5 | 5 | 3 | 3 |