# Generic classifications by p-group class

This page contains results for classes of *p*-groups for which we either have classifications or have general results concerning Morita equivalence classes.

## Contents

- 1 Fusion trivial
*p*-groups - 2 Cyclic
*p*-groups - 3 Tame blocks
- 4 Abelian [math]2[/math]-groups with [math]2[/math]-rank at most three
- 5 Minimal nonabelian [math]2[/math]-groups
- 6 Homocyclic [math]2[/math]-groups when inertial quotient contains a Singer cycle
- 7 Abelian [math]2[/math]-groups when inertial quotient is cyclic and acts freely on the defect group

## Fusion trivial *p*-groups

*p*-groups [math]P[/math] for which the only saturated fusion system is [math]\mathcal{F}_P(P)[/math] have not yet been given a name in the literature (to our knowledge). We will call them *fusion trivial*, but *nilpotent forcing* also seems appropriate following [vdW91] (where *p*-groups for which any finite group [math]G[/math] containing [math]P[/math] as a Sylow *p*-subgroup must be *p*-nilpotent are called *p-nilpotent forcing*). It is not known whether these two definitions are equivalent, i.e., whether there exist *p*-nilpotent forcing *p*-groups for which there is an exotic fusion system.

Blocks with fusion trivial defect groups must be nilpotent by [Pu88].

Examples of fusion trivial *p*-groups are abelian *2*-groups with automorphism group a *2*-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [CG12] or [Sa12b]).

Note that a *p*-group is fusion trivial if and only if it is resistant and has automorphism group a *p*-group. See [St06] for an analysis of resistant *p*-groups.

## Cyclic *p*-groups

Click here for background on blocks with cyclic defect groups.

Morita equivalence classes are labelled by Brauer trees, but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each *k*-Morita equivalence class corresponds to an unique [math]\mathcal{O}[/math]-Morita equivalence class.

For [math]p=2,3[/math] every appropriate Brauer tree is realised by a block and we can give generic descriptions.

[math]2[/math]-blocks with cyclic defect groups

[math]3[/math]-blocks with cyclic defect groups

## Tame blocks

Click here for background on tame blocks.

Erdmann classified algebras which are candidates for basic algebras of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [Er90] ) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field *k* at present.

Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [KoLa20].

## Abelian [math]2[/math]-groups with [math]2[/math]-rank at most three

These have been classified in [WZZ18] and [EL18a] with respect to [math]\mathcal{O}[/math]. The derived equivalences classes with respect to [math]\mathcal{O}[/math] are known.

Let [math]l,m,n \geq 1[/math] be distinct with [math]l,m \neq 1[/math]

[math]rk_2(D) \leq 3[/math]
| |||||

[math]D[/math] | # [math](\mathcal{O}[/math]-)Morita classes | # [math]\mathcal{O}[/math]-Derived equiv classes | Always endopermutation source Morita equivalent? | References | Notes |
---|---|---|---|---|---|

[math]C_{2^n}[/math] | 1 | 1 | Yes | [Li96] | Blocks with cyclic defect groups |

[math]C_2 \times C_2[/math] | 3 | 2 | Yes | [Li94] | |

[math]C_{2^m} \times C_{2^m}[/math] | 2 | 2 | Yes | [EKKS14] | |

[math]C_{2^m} \times C_{2^n}[/math] | 1 | 1 | Yes | [Pu88] | Nilpotent blocks |

[math]C_2 \times C_2 \times C_2[/math] | 8 | 4 | [Ea16] | ||

[math]C_{2^m} \times C_{2^m} \times C_{2^m}[/math] | 4 | 4 | Yes | [WZZ18] | |

[math]C_{2^m} \times C_2 \times C_2[/math] | 3 | 2 | [EL18a] | ||

[math]C_{2^m} \times C_{2^m} \times C_{2^n}[/math] | 2 | 2 | [WZZ18] | ||

[math]C_{2^l} \times C_{2^m} \times C_{2^n}[/math] | 1 | 1 | Yes | [Pu88] | Nilpotent blocks |

## Minimal nonabelian [math]2[/math]-groups

Blocks with defect groups which are minimal nonabelian [math]2[/math]-groups of the form [math]P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math] are classified in [EKS12]. There are two [math]\mathcal{O}[/math]-Morita equivalence classes, with representatives [math]\mathcal{O}P[/math] and [math]\mathcal{O}(P:C_3)[/math].

For arbitrary minimal nonabelian [math]2[/math]-groups, by [Sa16] blocks with such defect groups and the same fusion system are isotypic.

## Homocyclic [math]2[/math]-groups when inertial quotient contains a Singer cycle

A Singer cycle is an element of order [math]p^n-1[/math] in [math]\operatorname{Aut}({C_p}^n) \cong GL_n(p)[/math], and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of [math](C_p)^n[/math].

[math]2[/math]-blocks with homocyclic defect group [math] D \cong (C_{2^m})^n [/math] whose inertial quotient [math] \mathbb{E} [/math] contains a Singer cycle are classified in [McK19]. In this situation, [math]\mathbb{E}[/math] has the form [math]E:F [/math] where [math]E \cong C_{2^n-1}[/math] and [math]F[/math] is trivial or a subgroup of [math]C_n[/math]. There are three [math] \mathcal{O} [/math]-Morita equivalence classes when [math]m=1, n =3 [/math]; two when [math]m=1 , n \neq 3 [/math]; and only one when [math]m\gt 1 [/math]. The three classes have representatives [math] \mathcal{O}J_1 [/math], which occurs only when [math]m=1, n=3[/math]; [math] \mathcal{O}(SL_2(2^n):F) [/math], which occurs only when [math]m=1 [/math]; and [math] \mathcal{O}(D : \mathbb{E})[/math].

The Morita equivalence between the block and the class representative is known to be basic, possibly except when [math]m=1,n=3[/math], since the Morita equivalences between the principal block of [math]{\rm \operatorname{Aut}(SL_2(8))}[/math] and the blocks [math]B_0(\mathcal{O}({}^2G_2(3^{2m+1})))[/math] are not known to be basic.

## Abelian [math]2[/math]-groups when inertial quotient is cyclic and acts freely on the defect group

[math]2[/math]-blocks with abelian defect group [math] D [/math] whose inertial quotient [math] E [/math] is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in [math] E [/math] of any nontrivial element of [math]D[/math] is trivial) are classified in [ArMcK20].

If the action of [math]E[/math] is transitive, then [math]D[/math] is homocyclic, [math]E[/math] is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of [math] \mathcal{O}SL_2(2^n) [/math], or to [math] \mathcal{O}(D : E)[/math].

If the action of [math]E[/math] is not transitive, then the block is Morita equivalent to [math] \mathcal{O}(D : E)[/math].

In each case, the Morita equivalence between the block and the class representative is known to be basic.