Construction of 2–local finite groups of a type studied by Solomon and Benson

A $p$ –local finite group is an algebraic structure with a classifying space which has many of the properties of $p$ –completed classifying spaces of finite groups. In this paper, we construct a family of 2–local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2–subgroup of ${Spin}_{7} (q)$ ( $q$ an odd prime power) shown by Solomon not to occur as the 2–fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the $2$ –completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer–Wilkerson space $B D I (4)$ .

Keywords

Classifying space, $p$–completion, finite groups, fusion.

Mathematical Subject Classification 2000

Primary: 55R35

Secondary: 55R37, 20D06, 20D20

References

Forward citations

Publication

Received: 22 October 2002

Accepted: 31 December 2002

Published: 31 December 2002

Proposed: Haynes Miller

Seconded: Ralph Cohen, Bill Dwyer

Authors

Ran Levi
Department of Mathematical Sciences University of Aberdeen Meston Building 339 Aberdeen AB24 3UE United Kingdom

Bob Oliver
LAGA – UMR 7539 of the CNRS Institut Galilée Av J-B Clément 93430 Villetaneuse France

Volume 6, issue 2 (2002)

Ran Levi and Bob Oliver