See element structure of symmetric group:S3 for full details.
Element orders and conjugacy class structure
Review the conjugacy class structure:
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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:
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Review the multiplication table in one-line notation:
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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:
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Review the commutator operation:
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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:
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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 |
|