# 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]

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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 | -- the identity element | 1 | even; no | 1 | ||

2 + 1 + 1 | 2 (1 time), 1 (2 times) | one transposition (cycle of size two), two fixed points | , , , , , | 6 | , also | 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 | or | even; yes; no | 3 | |

4 | 4 (1 time) | one 4-cycle, no fixed points | , , , , , | 6 | or | 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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