Subgroup structure of symmetric group:S3
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This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S3.
View subgroup structure of particular groups | View other specific information about symmetric group:S3
Since this group is a complete group, every automorphism of it is inner, and in particular, this means that the classification of subgroups upto conjugacy is the same as the classification up to automorphism.
Contents
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group
Quick summary
Item | Value |
---|---|
Number of subgroups | 6 Compared with : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 4 Compared with : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 4 Compared with : 1,2,4,11,19,37,96,296,554,1593 |
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems | 2-Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one 3-Sylow: cyclic group:Z3, Sylow number is 1, fusion system is non-inner fusion system for cyclic group:Z3 |
Hall subgroups | Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a square-free number |
maximal subgroups | maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3). |
normal subgroups | There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3. |
Table classifying subgroups up to automorphisms
For more information on each automorphism type, follow the link.
Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Note |
---|---|---|---|---|---|---|---|---|---|
trivial subgroup | trivial group | 1 | 6 | 1 | 1 | 1 | symmetric group:S3 | trivial | |
S2 in S3 | cyclic group:Z2 | 2 | 3 | 1 | 3 | 3 | -- | 2-Sylow | |
A3 in S3 | cyclic group:Z3 | 3 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 3-Sylow | |
whole group | |
symmetric group:S3 | 6 | 1 | 1 | 1 | 1 | trivial group | |
Total (4 rows) | -- | -- | -- | -- | 4 | -- | 6 | -- | -- |
Table classifying isomorphism types of subgroups
Note that the first part of the GAP ID is the order.
Group name | GAP ID | Index | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|
Trivial group | 6 | 1 | 1 | 1 | 1 | |
Cyclic group:Z2 | 3 | 3 | 1 | 0 | 0 | |
Cyclic group:Z3 | 2 | 1 | 1 | 1 | 1 | |
Symmetric group:S3 | 1 | 1 | 1 | 1 | 1 | |
Total (4 rows) | -- | -- | 6 | 4 | 3 | 3 |
Table listing number of subgroups by order
Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order is congruent to modulo .
Group order | Index | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|
1 | 6 | 1 | 1 | 1 | 1 |
2 | 3 | 3 | 1 | 0 | 0 |
3 | 2 | 1 | 1 | 1 | 1 |
6 | 1 | 1 | 1 | 1 | 1 |
Total (4 rows) | -- | 6 | 4 | 3 | 3 |
Table listing numbers of subgroups by group property
Group property | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|
Cyclic group | 5 | 3 | 2 | 2 |
Abelian group | 5 | 3 | 2 | 2 |
Nilpotent group | 5 | 3 | 2 | 2 |
Solvable group | 6 | 4 | 3 | 3 |
Subgroup structure viewed as symmetric group
Classification based on partition given by orbit sizes
For any subgroup of , the natural action on induces a partition of the set into orbits, which in turn induces an unordered integer partition of the number 3. Below, we classify this information for the subgroups.
Conjugacy class of subgroups | Size of conjugacy class | Induced partition of 3 | Direct product of transitive subgroups on each orbit? | Illustration with representative |
---|---|---|---|---|
trivial subgroup | 1 | 1 + 1 + 1 | Yes | Under the action of the trivial subgroup, the orbits are singleton subsets. |
S2 in S3 | 3 | 2 + 1 | Yes | Under the action of , the orbits are and . |
A3 in S3 | 1 | 3 | Yes | The action is a transitive group action, so is a single orbit. Each of the elements sends the point to a different element of , covering everything. |
whole group | 1 | 3 | Yes | The action is a transitive group action, so is a single orbit. |
Defining functions
Subgroup-defining functions and associated quotient-defining functions
Subgroup series-defining functions
Series-defining function | Type | Zeroth member | First member | Second member | Third member | Stable member |
---|---|---|---|---|---|---|
upper central series | ascending | trivial | center: trivial | second center: trivial | trivial | trivial |
lower central series | descending | -- | whole group | derived subgroup: -- A3 in S3 | -- A3 in S3 | -- A3 in S3 |
derived series | descending | whole group | derived subgroup: -- A3 in S3 | second derived subgroup: trivial | trivial | trivial |
Frattini series | descending | whole group | Frattini subgroup: trivial | trivial | trivial | trivial |
Fitting series | ascending | trivial | Fitting subgroup: -- A3 in S3 | whole group | whole group | whole group |
socle series | ascending | trivial | socle: -- A3 in S3 | whole group | whole group | whole group |
Conjugacy class-defining functions
Conjugacy class-defining function | What it means | Value as subgroup | Value as group | Order | Index of subgroup | Number of subgroups |
---|---|---|---|---|---|---|
2-Sylow subgroup | subgroup whose order is a power of 2, index relatively prime to 2. Sylow subgroups exist, and Sylow implies order-conjugate | S2 in S3: , | cyclic group:Z2 | 2 | 3 | 3 |
3-Sylow subgroup | subgroup whose order is a power of 3, index relatively prime to 3. Sylow subgroups exist, and Sylow implies order-conjugate | A3 in S3: | cyclic group:Z3 | 3 | 2 | 1 |
Lattice of subgroups
The entire lattice
The lattice of subgroups of the symmetric group of degree three has the following interesting features:
- Every non-identity automorphism of the whole group acts nontrivially on the lattice. Note that since the symmetric group of degree three is a complete group, all the automorphisms are inner.
- In fact, the non-identity automorphisms give rise to all possible permutations of the three non-abelian subgroups of order two. More specifically, a permutation of the letters gives rise to an inner automorphism that permutes the two-element subgroups fixing these elements the same way. For instance, the -cycle , acting by conjugation, sends the subgroup stabilizing (namely ) to the subgroup stabilizing (namely ).
The sublattice of normal subgroups
The lattice of normal subgroups, which is in this case also the lattice of characteristic subgroups, is a totally ordered sublattice comprising the trivial subgroup, the subgroup of order three, and the whole group. This sublattice is preserved by all automorphisms.
Subgroup operations
Intersection
For all pairs of subgroups, either one is contained in the other, or the intersection is trivial. This makes the intersection table easy to construct.
Subgroup/subgroup | trivial | whole group | ||||
---|---|---|---|---|---|---|
trivial | trivial | trivial | trivial | trivial | trivial | trivial |
trivial | trivial | trivial | trivial | |||
trivial | trivial | trivial | trivial | |||
trivial | trivial | trivial | trivial | |||
trivial | trivial | trivial | trivial | |||
whole group | trivial | whole group |
Join of subgroups
For all pairs of subgroups, either one is contained in the other, or the join is the whole group. This makes the table for joins of subgroups easy to construct.
Subgroup/subgroup | trivial | whole group | ||||
---|---|---|---|---|---|---|
trivial | trivial | whole group | ||||
whole group | whole group | whole group | whole group | |||
whole group | whole group | whole group | whole group | |||
whole group | whole group | whole group | whole group | |||
whole group | whole group | whole group | whole group | |||
whole group | whole group | whole group | whole group | whole group | whole group | whole group |
Commutators of pairs of subgroups
For any pair of subgroups, their commutator is trivial if one of them is trivial or they are both equal and proper, and is otherwise.
Subgroup/subgroup | trivial | whole group | ||||
---|---|---|---|---|---|---|
trivial | trivial | trivial | trivial | trivial | trivial | trivial |
trivial | trivial | |||||
trivial | trivial | |||||
trivial | trivial | |||||
trivial | trivial | |||||
whole group | trivial |
Unary operations on subgroups
Subgroup | Normalizer | Centralizer | Normal closure | Normal core |
---|---|---|---|---|
trivial | whole group | whole group | trivial | trivial |
whole group | trivial | |||
whole group | trivial | |||
whole group | trivial | |||
whole group | ||||
whole group | whole group | trivial | whole group | whole group |
Subgroup series
The max-length of the group is 2 (it cannot be more, based on the prime factorization of 6) and there are four subgroup series of this maximal length, one series for each proper nontrivial subgroup.
Series | Intermediate subgroup | Normal series? | Subnormal series? | Chief series? | Composition series? |
---|---|---|---|---|---|
-- S2 in S3 | No | No | No | No | |
-- S2 in S3 | No | No | No | No | |
-- S2 in S3 | No | No | No | No | |
-- A3 in S3 | Yes | Yes | Yes | Yes |
In particular, there is a unique composition series which is also a unique chief series for the group.