Quiz:Subgroup structure of symmetric group:S3
From Groupprops
See subgroup structure of symmetric group:S3 for background information.
Basic stuff
Summary table on the structure of subgroups: [SHOW MORE]
See subgroup structure of symmetric group:S3 for background information.
Item | Value |
---|---|
Number of subgroups | 6 Compared with : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 4 Compared with : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 4 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: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one 3-Sylow: cyclic group:Z3, Sylow number is 1, 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. Interestingly, all subgroups are Hall subgroups, because the order is a square-free number |
maximal subgroups | maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3). |
normal subgroups | There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3. |
For more information on each automorphism type, follow the link.
Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Note |
---|---|---|---|---|---|---|---|---|---|
trivial subgroup | trivial group | 1 | 6 | 1 | 1 | 1 | symmetric group:S3 | trivial | |
S2 in S3 | cyclic group:Z2 | 2 | 3 | 1 | 3 | 3 | -- | 2-Sylow | |
A3 in S3 | cyclic group:Z3 | 3 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 3-Sylow | |
whole group | |
symmetric group:S3 | 6 | 1 | 1 | 1 | 1 | trivial group | |
Total (4 rows) | -- | -- | -- | -- | 4 | -- | 6 | -- | -- |