Classification by p-group

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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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