group extension

Let G and H be groups. A group E is called an extensionPlanetmathPlanetmathPlanetmathPlanetmath of G by H if

  1. 1.

    G is isomorphicPlanetmathPlanetmathPlanetmath to a normal subgroupMathworldPlanetmath N of E, and

  2. 2.

    H is isomorphic to the quotient groupMathworldPlanetmath E/N.

The definition is well-defined and it is convenient sometimes to regard G as a normal subgroup of E. The definition can be alternatively defined: E is an extension of G by H if there is a short exact sequenceMathworldPlanetmathPlanetmath of groups:


In fact, some authors define an extension (of a group by a group) to be a short exact sequence of groups described above. Also, many authors prefer the reverse terminology, calling the group E an extension of H by G.


  • Given any groups G and H, an extension of G by H exists: take the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of G and H.

  • An intermediate concept between an extension a direct product is that of a semidirect productMathworldPlanetmath of two groups: If G and H are groups, and E is an extension of G by H (identifying G with a normal subgroup of E), then E is called a semidirect product of G by H if

    1. (a)

      H is isomorphic to a subgroupMathworldPlanetmathPlanetmath of E, thus viewing H as a subgroup of E,

    2. (b)

      E=GH, and

    3. (c)


    Equivalently, E is a semidirect product of G and H if the short exact sequence


    splits. That is, there is a group homomorphism ϕ:HE such that the composition


    gives the identity map. Thus, a semidirect product is also known as a split extension. That a semidirect product E of G by H is also an extension of G by H can be seen via the isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath hhG.

    Furthermore, if H happens to be normal in E, then E is isomorphic to the direct product of G and H. (We need to show that (g,h)gh is an isomorphism. It is not hard to see that the map is a bijection. The trick is to show that it is a homomorphismMathworldPlanetmathPlanetmathPlanetmath, which boils down to showing that every element of G commutes with every element of H. To show the last step, suppose ghg-1=h¯H. Then gh=h¯g, so ghh¯-1=h¯gh¯-1=g¯G, or that hh¯-1=g-1g¯. Therefore, h=h¯.)

  • The extension problem in group theory is the classification of all extension groups of a given group G by a given group H. Specifically, it is a problem of finding all “inequivalent” extensions of G by H. Two extensions E1 and E2 of G by H are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if there is a homomorphism e:E1E2 such that the following diagram of two short exact sequences is commutativePlanetmathPlanetmathPlanetmath:


    According to the 5-lemma, e is actually an isomorphism. Thus equivalences of extensions are well-defined.

  • Like split extensions, special extensions are formed when certain conditions are imposed on G, H, or even E:

    1. (a)

      If all the groups involved are abelian (only that E is abelian is necessary here), then we have an abelian extensionMathworldPlanetmath.

    2. (b)

      If G, considered as a normal subgroup of E, actually lies within the center of E, then E is called a central extension. A central extension that is also a semidirect product is a direct product. Indeed, if E is both a central extension and a semidirect product of G by H, we observe that (gh¯)h(gh¯)-1=h¯hh¯-1H so that H is normal in E. Applying this result to the previous discussion and we have EG×H.

    3. (c)

      If G is a cyclic groupMathworldPlanetmath, then the extensions in question are called cyclic extensions.

Title group extension
Canonical name GroupExtension
Date of creation 2013-03-22 15:24:25
Last modified on 2013-03-22 15:24:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 20J05
Related topic HNNExtension
Defines split extension
Defines abelian extension
Defines central extension
Defines cyclic extension
Defines extension problem