Characteristic 2 type

In finite group theory, a branch of mathematics, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristic 2.

In the classification of finite simple groups, there is a major division between group of characteristic 2 type, where involutions resemble unipotent elements, and other groups, where involutions resemble semisimple elements.

Groups of characteristic 2 type and rank at least 3 are classified by the trichotomy theorem.

Definitions

A group is said to be of even characteristic if

$C_{M}(O_{2}(M))\leq O_{2}(M)$ for all maximal 2-local subgroups M that contain a Sylow 2-subgroup of G,

where $O_{2}(M)$ denotes the 2-core, the largest normal 2-subgroup (subgroup in which the order of every element is a power of 2) of M, which is the intersection of all conjugates of any given Sylow 2-subgroup. If this condition holds for all maximal 2-local subgroups M then G is said to be of characteristic 2 type. Gorenstein, Lyons & Solomon (1994, p.55) use a modified version of this called even type.

References

• Aschbacher, Michael; Smith, Stephen D. (2004), The classification of quasithin groups. I Structure of Strongly Quasithin K-groups, Mathematical Surveys and Monographs, vol. 111, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3410-7, MR 2097623
• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1994), The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0334-9, MR 1303592