See symmetric group:S3. We take the symmetric group on the set
of size three.
Elements
See element structure of symmetric group:S3 for full details.
Element orders and conjugacy class structure
Review the conjugacy class structure:
[SHOW MORE]
Partition |
Partition in grouped form |
Verbal description of cycle type |
Elements with the cycle type in cycle decomposition notation |
Elements with the cycle type in one-line notation |
Size of conjugacy class |
Formula for size |
Even or odd? If even, splits? If splits, real in alternating group? |
Element order |
Formula calculating element order
|
1 + 1 + 1 |
1 (3 times) |
three fixed points |
-- the identity element |
123 |
1 |
|
even; no |
1 |
|
2 + 1 |
2 (1 time), 1 (1 time) |
transposition in symmetric group:S3: one 2-cycle, one fixed point |
, , |
213, 321, 132 |
3 |
|
odd |
2 |
|
3 |
3 (1 time) |
3-cycle in symmetric group:S3: one 3-cycle |
, |
231, 312 |
2 |
|
even; yes; no |
3 |
|
Total (3 rows -- 3 being the number of unordered integer partitions of 3) |
-- |
-- |
-- |
-- |
6 (equals 3!, the size of the symmetric group) |
-- |
odd: 3 even;no: 1 even; yes; no: 2 |
order 1: 1, order 2: 3, order 3: 2 |
--
|
Multiplication, conjugacy and generating sets
Review the multiplication table in cycle decomposition notation:
[SHOW MORE]
Review the multiplication table in one-line notation:
[SHOW MORE]
Element |
123 |
213 |
132 |
321 |
231 |
312
|
123 |
123 |
213 |
132 |
321 |
231 |
312
|
213 |
213 |
123 |
231 |
312 |
132 |
321
|
132 |
132 |
312 |
123 |
231 |
321 |
213
|
321 |
321 |
231 |
312 |
123 |
213 |
132
|
231 |
231 |
321 |
213 |
132 |
312 |
123
|
312 |
312 |
132 |
321 |
213 |
123 |
231
|
Conjugation and commutator operations
Review the conjugation operation:
[SHOW MORE]
Review the commutator operation:
[SHOW MORE]
Here, the two inputs are group elements
, and the output is the commutator. We first give the table assuming the left definition of commutator:
. Here, the row element is
and the column element is
. Note that
:
The corresponding table with the right definition:
PLACEHOLDER FOR INFORMATION TO BE FILLED IN:
[SHOW MORE]One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the
error reporting page.
Here is the information on the number of times each element occurs as a commutator:
Conjugacy class (indexing partition) |
Elements |
Number of occurrences of each as commutator |
Probability of each occurring as the commutator of elements picked uniformly at random |
Total number of occurrences as commutator |
Total probability |
Explanation
|
1 + 1 + 1 |
|
18 |
1/2 |
18 |
1/2 |
See commuting fraction and its relationship with the number of conjugacy classes.
|
2 + 1 |
|
0 |
0 |
0 |
0 |
Not in the derived subgroup.
|
3 |
|
9 |
1/4 |
18 |
1/2 |
|
Subgroups
See subgroup structure of symmetric group:S3 for background information.
Basic stuff
Summary table on the structure of subgroups:
[SHOW MORE]
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.
View the lattice of subgroups as a picture:
[SHOW MORE]
Linear representations
For background, see linear representation theory of symmetric group:S3.
View overall summary:
[SHOW MORE]
Item |
Value
|
Degrees of irreducible representations over a splitting field (and in particular over ) |
1,1,2 maximum: 2, lcm: 2, number: 3 sum of squares: 6, quasirandom degree: 1
|
Schur index values of irreducible representations |
1,1,1
|
Smallest ring of realization for all irreducible representations (characteristic zero) |
|
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) |
(hence, it is a rational representation group)
|
Condition for being a splitting field for this group |
Any field of characteristic not two or three is a splitting field. In particular, and are splitting fields.
|
Minimal splitting field in characteristic |
The prime field
|
Smallest size splitting field |
field:F5, i.e., the field of five elements.
|
View representations summary:
[SHOW MORE]
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.
View character table:
[SHOW MORE]
Basic stuff
Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the one-dimensional trivial representation, the one-dimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a two-dimensional representation.