Quiz:Symmetric group:S4
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See symmetric group:S4. We take the symmetric group on the set of size four.
Elements
See element structure of symmetric group:S4 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 | 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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Subgroups
See subgroup structure of symmetric group:S4 for background information.
Basic stuff
Summary table on the structure of subgroups: [SHOW MORE]Quick summary
Item | Value |
---|---|
Number of subgroups | 30 Compared with : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 11 Compared with : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 11 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 3, fusion system is non-inner non-simple fusion system for dihedral group:D8 3-Sylow: cyclic group:Z3, Sylow number is 4, fusion system is non-inner fusion system for cyclic group:Z3 |
Hall subgroups | Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups |
maximal subgroups | maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4). |
normal subgroups | There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4. |
Table classifying subgroups up to automorphisms
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.