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)) \le 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 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, Vol. 111, Mathematical Surveys and Monographs, 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, Vol. 40, Mathematical Surveys and Monographs, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0334-9. MR 1303592