# Quiz:Element structure of symmetric group:S5

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See element structure of symmetric group:S5 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:S5 for full details.

Partition | Partition in grouped form | Verbal description of cycle type | Representative element with the cycle type | Size of conjugacy class | Formula calculating size | Even or odd? If even, splits? If splits, real in alternating group? | Element order | Formula calcuating element order |
---|---|---|---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 | 1 (5 times) | five fixed points | -- the identity element | 1 | even; no | 1 | ||

2 + 1 + 1 + 1 | 2 (1 time), 1 (3 times) | transposition: one 2-cycle, three fixed point | 10 | or , also in this case | odd | 2 | ||

3 + 1 + 1 | 3 (1 time), 1 (2 times) | one 3-cycle, two fixed points | 20 | or | even; no | 3 | ||

2 + 2 + 1 | 2 (2 times), 1 (1 time) | double transposition: two 2-cycles, one fixed point | 15 | or | even; no | 2 | ||

4 + 1 | 4 (1 time), 1 (1 time) | one 4-cycle, one fixed point | 30 | or | odd | 4 | ||

3 + 2 | 3 (1 time), 2 (1 time) | one 3-cycle, one 2-cycle | 20 | or | odd | 6 | ||

5 | 5 (1 time) | one 5-cycle | 24 | or | even; yes; yes | 5 | ||

Total (7 rows, 7 being the number of unordered integer partitions of 5) | -- | -- | -- | 120 (equals order of the group) | -- | odd: 60 (3 classes) even;no: 36 (3 classes) even;yes;yes: 24 (1 class) |
[SHOW MORE] order 1: 1 (1 class) order 2: 25 (2 classes) order 3: 20 (1 class) order 4: 30 (1 class) order 5: 24 (1 class) order 6: 20 (1 class) |

- This page was last edited on 19 October 2011, at 23:18.
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