In
finite group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite ...
over a
field of
characteristic 2.
In the
classification of finite simple groups
In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
, 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
:
for all maximal 2-local subgroups ''M'' that contain a
Sylow 2-subgroup
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...
of ''G'',
where
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.
use a modified version of this called even type.
References
*
*{{Citation , last1=Gorenstein , first1=D. , author1-link=Daniel Gorenstein , last2=Lyons , first2=Richard , last3=Solomon , first3=Ronald , title=The classification of the finite simple groups , url=https://www.ams.org/online_bks/surv401 , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, R.I. , series=Mathematical Surveys and Monographs , isbn=978-0-8218-0334-9 , mr=1303592 , year=1994 , volume=40
Finite groups