Fusion systems for groups of order 8: Difference between revisions
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==Information on number of fusion systems== | ==Information on number of saturated fusion systems== | ||
{{fusion systems facts to check against}} | {{fusion systems facts to check against}} | ||
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! Group !! GAP ID second part !! Hall-Senior number !! [[Nilpotency class]] !! Number of fusion systems (strict counting) !! Number of fusion systems up to isomorphism !! Does the identity functor control strong fusion? (Yes if abelian) | ! Group !! GAP ID second part !! Hall-Senior number !! [[Nilpotency class]] !! Number of saturated fusion systems (strict counting) !! Number of saturated fusion systems up to isomorphism !! Does the identity functor control strong fusion in ''every'' saturated fusion system? (Yes if abelian) | ||
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| [[cyclic group:Z8]] || 1 || 3 || 1 || 1 || 1 || Yes | | [[cyclic group:Z8]] || 1 || 3 || 1 || 1 || 1 || Yes | ||
Revision as of 23:48, 4 May 2012
This article gives specific information, namely, fusion systems, about a family of groups, namely: groups of order 8.
View fusion systems for group families | View fusion systems for groups of a particular order |View other specific information about groups of order 8
| Group | GAP ID second part | Hall-Senior number | Nilpotency class | Fusion systems page |
|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | fusion systems for cyclic group:Z8 |
| direct product of Z4 and Z2 | 2 | 2 | 1 | fusion systems for direct product of Z4 and Z2 |
| dihedral group:D8 | 3 | 4 | 2 | fusion systems for dihedral group:D8 |
| quaternion group | 4 | 5 | 2 | fusion systems for quaternion group |
| elementary abelian group:E8 | 5 | 1 | 1 | fusion systems for elementary abelian group:E8 |
Information on number of saturated fusion systems
FACTS TO CHECK AGAINST FOR FUSION SYSTEMS:
For an abelian group of prime power order: identity functor controls strong fusion for saturated fusion system on abelian group|classification of saturated fusion systems on abelian group of prime power order
| Group | GAP ID second part | Hall-Senior number | Nilpotency class | Number of saturated fusion systems (strict counting) | Number of saturated fusion systems up to isomorphism | Does the identity functor control strong fusion in every saturated fusion system? (Yes if abelian) |
|---|---|---|---|---|---|---|
| cyclic group:Z8 | 1 | 3 | 1 | 1 | 1 | Yes |
| direct product of Z4 and Z2 | 2 | 2 | 1 | 1 | 1 | Yes |
| dihedral group:D8 | 3 | 4 | 2 | 4 | 3 | No |
| quaternion group | 4 | 5 | 2 | 2 | 2 | Yes |
| elementary abelian group:E8 | 5 | 1 | 1 | 45 | 4 | Yes |
| Total (5 groups) | -- | -- | -- | 53 | 11 | 4 Yes, 1 No |