Difference between revisions of "Classification by p-group"

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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.
 
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.
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== Blocks for <math> p=2 </math> ==
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{| 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> || ||
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|}
 +
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong>Defect group of size <math>2</math> &nbsp;</strong>
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|-
 +
! 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
 +
|-
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| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
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| <strong>Defect group of size <math>4</math> &nbsp;</strong>
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|-
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! 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
 +
|-
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| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|-
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| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
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| <strong>Defect group of size <math>8</math> &nbsp;</strong>
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|-
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! 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
 +
|-
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|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|-
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|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|-
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|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || || [[References|[Er87] ]] ||
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|-
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|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] || || <math>k</math> || || ||
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|-
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|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
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
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| <strong>Defect group of size <math>16</math> &nbsp;</strong>
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|-
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! 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
 +
|-
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|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|-
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|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||
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|-
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|16 || [[SmallGroup(16,3)|3]] || [[SmallGroup(16,3)]] || || || || [[References|[Sa11] ]] || Block invariants known
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|-
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|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || || || || [[References|[Sa12] ]] || Block invariants known
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|-
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|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|-
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|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || || || || [[References|[Sa12b] ]] || Block invariants known
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|-
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|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || || || || ||
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|-
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|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || || || || ||
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|-
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|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || || || || ||
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|-
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|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] ]] ||
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|-
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|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || || || || [[References|[Sa12] ]] || Block invariants known
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|-
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|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || || || || [[References|[Sa13] ]] || Block invariants known
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|-
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|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || || || || [[References|[Sa13b] ]] || Block invariants known
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|-
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|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||
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|}
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==Blocks for <math>p=3</math>==
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
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| <strong>Defect group of size <math>1</math> &nbsp;</strong>
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|-
 +
! 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
 +
|-
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| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
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| <strong>Defect group of size <math>3</math> &nbsp;</strong>
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|-
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! 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
 +
|-
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| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
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| <strong>Defect group of size <math>9</math> &nbsp;</strong>
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|-
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! 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
 +
|-
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|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|-
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|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||
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|}
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==Blocks for <math>p=5</math>==
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{| 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> || ||
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong>Defect group of size <math>5</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
 +
|-
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|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
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|}
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{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong>Defect group of size <math>25</math> &nbsp;</strong>
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|-
 +
! 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
 +
|-
 +
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 5(5) || No || <math>\mathcal{O}</math> || || Max 12 classes
 +
|-
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|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||
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|}
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==Blocks for <math>p=7</math>==
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{| 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> || ||
 +
|}
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{| 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
 +
|-
 +
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 19 classes
 +
|}
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 +
  
 
{| class="wikitable"
 
{| class="wikitable"

Revision as of 16:06, 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]


[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]
2 1 [math]C_2[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
3 1 [math]C_3[/math] 2(2) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
4 1 [math]C_4[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
4 2 [math]C_2 \times C_2[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Er82], [Li94]
5 1 [math]C_5[/math] 6(6) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
7 1 [math]C_7[/math] 14(14) No [math]\mathcal{O}[/math] Max 19 classes
8 1 [math]C_8[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
8 2 [math]C_4 \times C_2[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
8 3 [math]D_8[/math] 6(?) [math]k[/math] [Er87]
8 4 [math]Q_8[/math] [math]k[/math]
8 5 [math]C_2 \times C_2 \times C_2[/math] 8(8) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Ea16] Uses CFSG
9 1 [math]C_9[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
9 2 [math]C_3 \times C_3[/math]
11 1 [math]C_{11}[/math] No [math]\mathcal{O}[/math]
13 1 [math]C_{13}[/math] No [math]\mathcal{O}[/math]
16 1 [math]C_{16}[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
16 2 [math]C_4 \times C_4[/math] 2(2) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EKKS14]
16 3 SmallGroup(16,3) [Sa11] Block invariants known
16 4 [math]C_4:C_4[/math] [Sa12] Block invariants known
16 5 [math]C_8 \times C_2[/math] 1(1) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math]
16 6 [math]M_{16}[/math] [Sa12b] Block invariants known
16 7 [math]D_{16}[/math]
16 8 [math]SD_{16}[/math]
16 9 [math]Q_{16}[/math]
16 10 [math]C_4 \times C_2 \times C_2[/math] 3(3) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [EL18a]
16 11 [math]D_8 \times C_2[/math] [Sa12] Block invariants known
16 12 [math]Q_8 \times C_2[/math] [Sa13] Block invariants known
16 13 [math]D_8*C_4[/math] [Sa13b] Block invariants known
16 14 [math](C_2)^4[/math] 16(16) [math]\mathcal{O}[/math] [math]\mathcal{O}[/math] [Ea18]
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]
25 1 [math]C_{25}[/math] 5(5) No [math]\mathcal{O}[/math] Max 12 classes
25 2 [math]C_5 \times C_5[/math]
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