# Fusion systems for elementary abelian group:E8

From Groupprops

This article gives specific information, namely, fusion systems, about a particular group, namely: elementary abelian group:E8.

View fusion systems for particular groups | View other specific information about elementary abelian group:E8

## Summary

Item | Value |
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Total number of saturated fusion systems on a concrete instance of the group (strict, not up to isomorphism of fusion systems) |
45 By classification of saturated fusion systems on abelian group of prime power order, this equals the number of subgroups of odd order in the automorphism group of E8, which is GL(3,2). |

Total number of saturated fusion systems up to isomorphism | 4 By classification of saturated fusion systems on abelian group of prime power order, this equals the number of conjugacy classes of subgroups of odd order in the automorphism group of E8, which is GL(3,2). |

List of saturated fusion systems with orbit sizes | inner fusion system (orbit size 1 under isomorphisms), fusion system obtained from inside direct product of A4 and Z2 (orbit size 28 under isomorphisms), fusion system obtained from inside GA(1,8) (orbit size 8 under isomorphisms), fusion system obtained from inside the Ree group:Ree(3) (orbit size 8(?) under isomorphisms) |

Number of simple fusion systems | Up to isomorphism: 1 or 2 (not sure yet!) |

Number of maximal saturated fusion systems, i.e., saturated fusion systems not contained in bigger saturated fusion systems | 1 (the fusion system from Ree group:Ree(3)) |

## Description 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

Note that because the group is abelian, the classification of saturated fusion systems establishes a correspondence between isomorphism types of saturated fusion systems and conjugacy classes of odd-order subgroups of the automorphism group, which is GL(3,2), isomorphic to PSL(3,2). See also subgroup structure of projective special linear group:PSL(3,2).

Isomorphism type of fusion system | Number of such fusion systems under strict counting | Can the fusion system be realized using a Sylow subgroup of a finite group? | Does the identity functor control strong fusion? This would mean that all fusion occurs in the normalizer | Is the fusion system simple? | Smallest size embedding realizing this fusion system (if any) |
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inner fusion system, which in this case contains only subgroup inclusions since the group is abelian | 1 | Yes | Yes | No | as a subgroup of itself |

fusion system obtained by taking a semidirect product with cyclic group:Z3 that acts nontrivially on a two-dimensional subspace and fixes a one-dimensional complement. | 28 | Yes | Yes | No | as a subgroup inside direct product of A4 and Z2 |

fusion system obtained by taking a semidirect product with cyclic group:Z7 that permutes the seven non-identity elements | 8 | Yes | Yes | Yes | as a subgroup inside general affine group:GA(1,7) |

fusion system obtained by taking a semidirect product with a subgroup of order 21 in the automorphism group | 8 | Yes | Yes | Probably? | The smallest group has order 168. |