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

[math]\displaystyle{ C_M(O_2(M)) \le O_2(M) }[/math] for all maximal 2-local subgroups M that contain a Sylow 2-subgroup of G,

where [math]\displaystyle{ O_2(M) }[/math] denotes the 2-core, the largest normal 2-subgroup 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) 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, 111, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3410-7, https://www.ams.org/bookstore-getitem/item=SURV-111 
• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1994), The classification of the finite simple groups, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0334-9, https://www.ams.org/online_bks/surv401