Extension problem - Citizendia

In group theory, if the factor group G/K is isomorphic to H, one says that G is an extension of H by K. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group.

To consider some examples, if G = H × K, then G is an extension of both H and K. More generally, if G is a semidirect product of K and H, then G is an extension of H by K, so such products as the wreath product provide further examples of extensions. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, the wreath product of Group theory is a specialized product of two groups based on a Semidirect product.

The question of what groups G are extensions of H is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {A_i}, where each A_i+1 is an extension of A_i by some simple group. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning The classification of finite simple groups would give us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.

We can use the language of diagrams to provide a more flexible definition of extension: a group G is an extension of a group H by a group K if and only if there is an exact sequence:

$1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1$

where 1 denotes the trivial group with a single element. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group This definition is more general in that it does not require that K be a subgroup of G; instead, K is isomorphic to a normal subgroup K^* of G, and H is isomorphic to G/K^*. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

Classifying extensions

Solving the extension problem amounts to classifying all extensions of H by K; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition.

Classifying split extensions

A split extension is an extension

$1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1$

such that there is a homomorphism $s\colon H \rightarrow G$ such that going from H to G by s and then back to H by the quotient map induces the identity map on H. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector This article is about the Identity Map software design pattern In this situation, it is usually said that s splits the above exact sequence. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group

Split extensions are very easy to classify, because the splitting lemma states that an extension is split if and only if the group G is a semidirect product of K and H. In Mathematics, and more specifically in Homological algebra, the splitting lemma states that in any Abelian category, the following statements for ↔ In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from $H\to\operatorname{Aut}(K)$ , where Aut(K) is the automorphism group of K. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself For a full discussion of why this is true, see semidirect product. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can

Classifying extensions

Classifying split extensions

See also