# Alperin–Brauer–Gorenstein theorem

Jump to navigation
Jump to search

In mathematics, the **Alperin–Brauer–Gorenstein theorem** characterizes the finite simple groups with quasidihedral or wreathed^{[1]} Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite fields of odd order, depending on a certain congruence, or to the Mathieu group . Template:Harvtxt proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in Template:Harvtxt, and presented in some detail in Template:Harvtxt.

## Notes

- ↑ A 2-group is
**wreathed**if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyclic groups of the same order, that is, if it is the wreath product of a cyclic 2-group with the symmetric group on 2 points.

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}