Difference between revisions of "Classification by p-group"

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Line 171: Line 171:
 
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||  
 
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||  
 
|}
 
|}
−
 
  
 
==Blocks for <math>p=5</math>==
 
==Blocks for <math>p=5</math>==
Line 252: Line 251:
 
|}
 
|}
  
 +
==Blocks for <math>p=11</math>==
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong>Defect group of size <math>1</math> &nbsp;</strong>
 +
|-
 +
! scope="col"| <math>|D|</math>
 +
! scope="col"| SmallGroup
 +
! scope="col"| Isotype
 +
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes
 +
! scope="col"| Complete (w.r.t.)?
 +
! scope="col"| Derived equiv classes (w.r.t)?
 +
! scope="col"| References
 +
! scope="col"| Notes
 +
|-
 +
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 +
|}
 +
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong>Defect group of size <math>7</math> &nbsp;</strong>
 +
|-
 +
! scope="col"| <math>|D|</math>
 +
! scope="col"| SmallGroup
 +
! scope="col"| Isotype
 +
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes
 +
! scope="col"| Complete (w.r.t.)?
 +
! scope="col"| Derived equiv classes (w.r.t)?
 +
! scope="col"| References
 +
! scope="col"| Notes
 +
|-
 +
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 +
|}
  
 +
==Blocks for <math>p\geq 13</math>==
  
 
{| class="wikitable"
 
{| class="wikitable"
Line 266: Line 296:
 
|-  
 
|-  
 
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||  
 
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||  
−
|-
 
−
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||
 
−
|-
 
−
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 19 classes
 
−
|-
 
−
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || || [[References|[Er87] ]] ||
 
−
|-
 
−
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] || || <math>k</math> || || ||
 
−
|-
 
−
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG
 
−
|-
 
−
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||
 
−
|-
 
−
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
 
|-  
 
|-  
 
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||
−
|-
 
−
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||
 
−
|-
 
−
|16 || [[SmallGroup(16,3)|3]] || [[SmallGroup(16,3)]] || || || || [[References|[Sa11] ]] || Block invariants known
 
−
|-
 
−
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || || || || [[References|[Sa12] ]] || Block invariants known
 
−
|-
 
−
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
−
|-
 
−
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || || || || [[References|[Sa12b] ]] || Block invariants known
 
−
|-
 
−
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || || || || ||
 
−
|-
 
−
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || || || || ||
 
−
|-
 
−
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || || || || ||
 
−
|-
 
−
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a] ]] ||
 
−
|-
 
−
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || || || || [[References|[Sa12] ]] || Block invariants known
 
−
|-
 
−
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || || || || [[References|[Sa13] ]] || Block invariants known
 
−
|-
 
−
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || || || || [[References|[Sa13b] ]] || Block invariants known
 
−
|-
 
−
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||
 
 
|-
 
|-
 
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||
Line 330: Line 304:
 
|-
 
|-
 
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||
−
|-
 
−
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 5(5) || No || <math>\mathcal{O}</math> || || Max 12 classes
 
−
|-
 
−
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||
 
 
|}
 
|}

Revision as of 17:09, 28 August 2018

Classification of Morita equivalences for blocks with a given defect group

On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. Information on broad classes of p-groups can be found here.

We use the following notation for Morita equivalence classes of blocks of finite groups with respect to an algebraically closed field k.

[math]M(x,y,z)[/math] is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.

Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that [math]M(x,y1,z1)=M(x,y2,z2)[/math] for some [math](y1,z1) \neq (y2,z2)[/math].

Also, at present there is no known example of a k-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to the d.v.r.


Blocks for [math] p=2 [/math]

Blocks for [math]p=3[/math]

Blocks for [math]p=5[/math]

Blocks for [math]p=7[/math]

Blocks for [math]p=11[/math]

Blocks for [math]p\geq 13[/math]

[math]|D|[/math] SmallGroup Isotype Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes Complete (w.r.t.)? Derived equiv classes (w.r.t)? References Notes
1 1 [math]1[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
13 1 [math]C_{13}[/math] No [math]\mathcal{O}[/math]
17 1 [math]C_{17}[/math] No [math]\mathcal{O}[/math]
19 1 [math]C_{19}[/math] No [math]\mathcal{O}[/math]
23 1 [math]C_{23}[/math] No [math]\mathcal{O}[/math]
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