|
Balance Theorem
In mathematical group theory, the balance theorem states that if ''G'' is a group with no core then ''G'' either has disconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type . The significance of this theorem is that it splits the classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ... into three major subcases. References *{{Citation , last1=Gorenstein , first1=D. , author1-link=Daniel Gorenstein , title=The classification of finite simple groups. Vol. 1 , publisher=Plenum Press , series=The University Series in Mathematics , isbn=978-0-306-41305-6 , mr=746470 , year=1983 Theorems about finite groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core (group)
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group. The normal core Definition For a group ''G'', the normal core or normal interiorRobinson (1996) p.16 of a subgroup ''H'' is the largest normal subgroup of ''G'' that is contained in ''H'' (or equivalently, the intersection of the conjugates of ''H''). More generally, the core of ''H'' with respect to a subset ''S'' ⊆ ''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e. :\mathrm_S(H) := \bigcap_. Under this more general definition, the normal core is the core with respect to ''S'' = ''G''. The normal core of any normal subgroup is the subgroup itself. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylow 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 order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow ''p''-subgroup (sometimes ''p''-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a ''p''-group (meaning its cardinality is a power of p, or equivalently, the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written \text_p(G). The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G the order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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)) \le O_2(M) for all maximal 2-local subgroups ''M'' that contain a Sylow 2-subgroup of ''G''. 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=So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Finite Simple Groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
