Quiz:Symmetric group:S5
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See symmetric group:S5. We take the symmetric group on the set of size five.
Elements
See element structure of symmetric group:S5 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 | 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) |
Subgroups
See subgroup structure of symmetric group:S5 for background information and more details.
Basic stuff
Summary table on the structure of subgroups: [SHOW MORE]Quick summary
Item | Value |
---|---|
Number of subgroups | 156 Compared with : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 19 Compared with , : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 19 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: dihedral group:D8 (order 8), Sylow number is 15, fusion system is non-inner non-simple fusion system for dihedral group:D8 3-Sylow: cyclic group:Z3, Sylow number is 10, fusion system is non-inner fusion system for cyclic group:Z3 5-Sylow: Z5 in S5, Sylow number is 6, fusion system is universal fusion system for cyclic group:Z5 |
Hall subgroups | -Hall subgroup: S4 in S5 (order 24) No -Hall subgroup or -Hall subgroup |
maximal subgroups | maximal subgroups have orders 12 (direct product of S3 and S2 in S5), 20 (GA(1,5) in S5), 24 (S4 in S5), 60 (A5 in S5) |
normal subgroups | There are three normal subgroups: the whole group, A5 in S5, and the trivial subgroup. |
Table classifying subgroups up to automorphisms
Note that the only normal subgroups are the trivial subgroup, the whole group, and A5 in S5, so we do not waste a column on specifying whether the subgroup is normal and on the quotient group.
TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.