Quiz:Element structure of symmetric group:S4
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See element structure of symmetric group:S4 for full details.
Element orders and conjugacy class structure
Review the conjugacy class structure: [SHOW MORE]
See element structure of symmetric group:S4 for full details.
Partition | Partition in grouped form | Verbal description of cycle type | Elements with the cycle type | 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 | 1 (4 times) | four cycles of size one each, i.e., four fixed points | ![]() |
1 | ![]() |
even; no | 1 | ![]() |
2 + 1 + 1 | 2 (1 time), 1 (2 times) | one transposition (cycle of size two), two fixed points | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | ![]() ![]() |
odd | 2 | ![]() |
2 + 2 | 2 (2 times) | double transposition: two cycles of size two | ![]() ![]() ![]() |
3 | ![]() |
even; no | 2 | ![]() |
3 + 1 | 3 (1 time), 1 (1 time) | one 3-cycle, one fixed point | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | ![]() ![]() |
even; yes; no | 3 | ![]() |
4 | 4 (1 time) | one 4-cycle, no fixed points | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | ![]() ![]() |
odd | 4 | ![]() |
Total (5 rows, 5 being the number of unordered integer partitions of 4) | -- | -- | -- | 24 (equals 4!, the order of the whole group) | -- | odd: 12 (2 classes) even; no: 4 (2 classes) even; yes; no: 8 (1 class) |
order 1: 1 (1 class) order 2: 9 (2 classes) order 3: 8 (1 class) order 4: 6 (1 class) |
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