Subgroup structure of Mathieu group:M9: Difference between revisions
No edit summary |
|||
| Line 11: | Line 11: | ||
{| class="sortable" border="1" | {| class="sortable" border="1" | ||
! Automorphism class of subgroups !! List of 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) !! Subnormal depth (if subnormal) !! [[Nilpotency class]] (if nilpotent) | ! Automorphism class of subgroups !! List of 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) !! Subnormal depth (if subnormal) !! [[Nilpotency class]] (if nilpotent) !! Note | ||
|- | |- | ||
| trivial subgroup || || [[trivial group]] || 1 || 72 || 1 || 1 || 1 || [[Mathieu group:M9]] || 1 || 0 | | trivial subgroup || || [[trivial group]] || 1 || 72 || 1 || 1 || 1 || [[Mathieu group:M9]] || 1 || 0 || trivial | ||
|- | |- | ||
| || || [[cyclic group:Z2]] || 2 || 36 || 1 || 9 || 9 || -- || -- || 1 | | || || [[cyclic group:Z2]] || 2 || 36 || 1 || 9 || 9 || -- || -- || 1 || | ||
|- | |- | ||
| || || [[cyclic group:Z4]] || 4 || 18 || 3 || 9 || 27 || -- || -- || 1 | | || || [[cyclic group:Z4]] || 4 || 18 || 3 || 9 || 27 || -- || -- || 1 || | ||
|- | |- | ||
| || || [[quaternion group]] || 8 || 9 || 1 || 9 || 9 || -- || -- || 2 | | || || [[quaternion group]] || 8 || 9 || 1 || 9 || 9 || -- || -- || 2 || 2-Sylow | ||
|- | |- | ||
| || || [[cyclic group:Z3]] || 3 || 24 || 1 || 4 || 4 || -- || 2 || 1 | | || || [[cyclic group:Z3]] || 3 || 24 || 1 || 4 || 4 || -- || 2 || 1 || | ||
|- | |- | ||
| [[3-Sylow subgroup of Mathieu group:M9]] || || [[elementary abelian group:E9]] || 9 || 8 || 1 || 1 || 1 || [[quaternion group]] || 1 || 1 | | [[3-Sylow subgroup of Mathieu group:M9]] || || [[elementary abelian group:E9]] || 9 || 8 || 1 || 1 || 1 || [[quaternion group]] || 1 || 1 || 3-Sylow | ||
|- | |- | ||
| || || [[symmetric group:S3]] || 6 || 12 || 1 || 12 || 12 || -- || -- || -- | | || || [[symmetric group:S3]] || 6 || 12 || 1 || 12 || 12 || -- || -- || -- || | ||
|- | |- | ||
| || || [[generalized dihedral group for E9]] || 18 || 4 || 1 || 1 || 1 || [[Klein four-group]] || 1 || -- | | || || [[generalized dihedral group for E9]] || 18 || 4 || 1 || 1 || 1 || [[Klein four-group]] || 1 || -- || | ||
|- | |- | ||
| || || [[SmallGroup(36,9)]] || 36 || 2 || 3 || 1 || 3 || [[cyclic group:Z2]] || 1 || -- | | || || [[SmallGroup(36,9)]] || 36 || 2 || 3 || 1 || 3 || [[cyclic group:Z2]] || 1 || -- || | ||
|- | |- | ||
| whole group || || [[Mathieu group:M9]] || 72 || 1 || 1 || 1 || 1 || [[trivial group]] || 0 || -- | | whole group || || [[Mathieu group:M9]] || 72 || 1 || 1 || 1 || 1 || [[trivial group]] || 0 || -- || | ||
|- | |- | ||
| Total (10 rows) || -- || -- || -- || -- || 14 || -- || 68 || -- || -- || -- | | Total (10 rows) || -- || -- || -- || -- || 14 || -- || 68 || -- || -- || -- || | ||
|} | |} | ||
Revision as of 18:11, 12 July 2011
This article gives specific information, namely, subgroup structure, about a particular group, namely: Mathieu group:M9.
View subgroup structure of particular groups | View other specific information about Mathieu group:M9
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group
Table classifying subgroups up to automorphisms
| Automorphism class of subgroups | List of 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) | Subnormal depth (if subnormal) | Nilpotency class (if nilpotent) | Note |
|---|---|---|---|---|---|---|---|---|---|---|---|
| trivial subgroup | trivial group | 1 | 72 | 1 | 1 | 1 | Mathieu group:M9 | 1 | 0 | trivial | |
| cyclic group:Z2 | 2 | 36 | 1 | 9 | 9 | -- | -- | 1 | |||
| cyclic group:Z4 | 4 | 18 | 3 | 9 | 27 | -- | -- | 1 | |||
| quaternion group | 8 | 9 | 1 | 9 | 9 | -- | -- | 2 | 2-Sylow | ||
| cyclic group:Z3 | 3 | 24 | 1 | 4 | 4 | -- | 2 | 1 | |||
| 3-Sylow subgroup of Mathieu group:M9 | elementary abelian group:E9 | 9 | 8 | 1 | 1 | 1 | quaternion group | 1 | 1 | 3-Sylow | |
| symmetric group:S3 | 6 | 12 | 1 | 12 | 12 | -- | -- | -- | |||
| generalized dihedral group for E9 | 18 | 4 | 1 | 1 | 1 | Klein four-group | 1 | -- | |||
| SmallGroup(36,9) | 36 | 2 | 3 | 1 | 3 | cyclic group:Z2 | 1 | -- | |||
| whole group | Mathieu group:M9 | 72 | 1 | 1 | 1 | 1 | trivial group | 0 | -- | ||
| Total (10 rows) | -- | -- | -- | -- | 14 | -- | 68 | -- | -- | -- |