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The Robson classification, also known as the 10-groups classification or ten groups classification system (TGCS), is a system for classifying pregnant women who undergo childbirth. It was developed to allow more accurate comparison of caesarean section rates between different settings, whether they be individual hospitals or entire regions or ...
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...
If a group H is a homomorphic image (or a quotient group) of a group G then rank(H) ≤ rank(G). If G is a finite non-abelian simple group (e.g. G = A n, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups.
A class of groups is a set-theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection. . This concept arose from the necessity to work with a bunch of groups satisfying certain special property (for example finiteness or commutativit
I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: Grp → Set has a left adjoint given by the composite KF: Set → Mon → Grp , where F is the free functor ; this functor assigns to every ...
The simple N-groups were classified by Thompson (1968, 1970, 1971, 1973, 1974, 1974b) in a series of 6 papers totaling about 400 pages.The simple N-groups consist of the special linear groups PSL 2 (q), PSL 3 (3), the Suzuki groups Sz(2 2n+1), the unitary group U 3 (3), the alternating group A 7, the Mathieu group M 11, and the Tits group.
An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a . A non-finitely generated countable example is given by the additive group of the polynomial ring Z [ X ] {\displaystyle \mathbb {Z} [X]} (the free abelian group of countable rank).
In particular, the multiplicative group G m is the group GL(1), and so its group G m (k) of k-rational points is the group k* of nonzero elements of k under multiplication. Another reductive group is the special linear group SL ( n ) over a field k , the subgroup of matrices with determinant 1.