- Group extension
In

mathematics , a**group extension**is a general means of describing a group in terms of a particularnormal subgroup andquotient group . If "Q" and "N" are two groups, then "G" is an**extension**of "Q" by "N" if there is ashort exact sequence :$1\; ightarrow\; N\; ightarrow\; G\; ightarrow\; Q\; ightarrow\; 1.\; ,!$

If "G" is an extension of "Q" by "N", then "G" is a group, "N" is a

normal subgroup of "G" and thequotient group "G"/"N" isisomorphic to group "Q". Group extensions arise in the context of the**extension problem**, where the groups "Q" and "N" are known and the properties of "G" are to be determined.An extension is called a

**central extension**if the subgroup "N" lies in the center of "G".**Extensions in general**One extension, the

direct product , is immediately obvious. If one requires "G" and "Q" to beabelian group s, then the set of isomorphism classes of extensions of "Q" by a given (abelian) group "N" is in fact a group, which isisomorphic to $operatorname\{Ext\}^1\_\{mathbb\; Z\}(Q,N)$; cf. theExt functor . Several other general classes of extensions are known but no theory exists which treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the**extension problem**.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 thewreath product provide further examples of extensions.**Extension problem**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 somesimple group . Theclassification 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.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 ::$1\; ightarrow\; K\; ightarrow\; G\; ightarrow\; H\; ightarrow\; 1$

where 1 denotes the trivial group with a single element. This definition is more general in that it does not require that "K" be a subgroup of "G"; instead, "K" is

isomorphic to anormal subgroup "K"^{*}of "G", and "H" is isomorphic to "G"/"K"^{*}.**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:$1\; ightarrow\; K\; ightarrow\; G\; ightarrow\; H\; ightarrow\; 1$

such that there is a

homomorphism $scolon\; H\; ightarrow\; G$ such that going from "H" to "G" by "s" and then back to "H" by the quotient map induces theidentity map on "H". In this situation, it is usually said that "s"**splits**the aboveexact sequence .Split extensions are very easy to classify, because the

splitting lemma states that an extension is splitif and only if the group "G" is asemidirect product of "K" and "H". Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from $H\; ooperatorname\{Aut\}(K)$, where Aut("K") is theautomorphism group of "K". For a full discussion of why this is true, seesemidirect product .**Central extension**A

**central extension**of a group "G" is a short exact sequence of groups:$1\; ightarrow\; A\; ightarrow\; E\; ightarrow\; G\; ightarrow\; 1$such that "A" is in "Z"("E"), the center of the group E. The set of isomorphism classes of central extensions of "G" by "A" is in one-to-one correspondence with the cohomology group "H"^{2}"(G,A)", where the action of "G" on "A" is trivial.Examples of central extensions can be constructed by taking any group "G" and any

abelian group "A", and setting "E" to be "A"×"G". This kind of "split" example (asplit extension in the sense of theextension problem , since "G" is present as a subgroup of "E") isn't of particular interest. More serious examples are found in the theory ofprojective representation s, in cases where the projective representation cannot be lifted to an ordinarylinear representation .In the case of finite perfect groups, there is a

universal perfect central extension .Similarly, the central extension of a

Lie algebra is an exact sequence:$0\; ightarrow\; mathfrak\{a\}\; ightarrowmathfrak\{e\}\; ightarrowmathfrak\{g\}\; ightarrow\; 0$such that $mathfrak\{a\}$ is in the center of $mathfrak\{e\}$.**Lie groups**Central extensions of

Lie group s are identical tocovering space s of Lie groups.If the group "G" is a

Lie group , then a central extension of "G" is a Lie group as well, and the Lie algebra of a central extension of "G" is a central extension of the Lie algebra of "G". In the terminology oftheoretical physics , the generators of "E" not included in "G" are calledcentral charge s. These generators are in the center of the Lie algebra of "E"; byNoether's theorem , generators of symmetry groups correspond to conserved quantities, referred to as charges.In Lie group theory, central extensions arise in connection with

algebraic topology . Suppose "G" is a connected Lie group that is notsimply connected . Itsuniversal cover "G"* is again a Lie group, in such a way that the projection:π: "G"* → "G"

is a group homomorphism, and surjective. Its kernel is (up to isomorphism) the

fundamental group of "G"; this is known to be abelian (seeH-space ). This construction gives rise to central extensions.Conversely, given a Lie group "G", with non-trivial center "Z", the quotient "G"/"Z" is a Lie group and "G" is a central extension of it.

The basic examples are:

* thespin group s, which double cover thespecial orthogonal group s, which (in even dimension) double-cover theprojective orthogonal group .

* themetaplectic group s, which double cover thesymplectic group s.The case of "SL"_{2}("R") involves a fundamental group that isinfinite cyclic . Here the central extension involved is well known inmodular form theory, in the case of forms of weight ½. A projective representation that corresponds is theWeil representation , constructed from theFourier transform , in this case on thereal line . Metaplectic groups also occur inquantum mechanics .**ee also***

Algebraic extension

*Field extension

*Ring extension

*Group cohomology

*Virasoro algebra

*HNN extension **References**

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