# Extensions of Groups

Whenever a group G has a normal subgroup N, we may say that G is an extension   of  N   by some group H which is isomorphic to the quotient G/N.   This page aims to describe the four ways in which an extension can work. It uses non-standard terminology.

The four ways are:

Whichever method we use, the size of G is the product of the sizes of N and H.

• ## Direct Products

Formally, we can regard each element of G as an ordered pair of elements of N,H. The group operation of G=N×H is ( na, hp ) * ( nb, hq ) = ( na*nb, hp*hq ).

Intuitively, the two component groups N and H do not interact. They just sit there side by side making a larger group. H is a also normal subgroup of G. Every element of the form (na, 1) commutes with every element of the form (1, hp). The direct product of N and H is the same as the direct product of H and N, i.e. N×H ≅ H×N.

• ## Semidirect Products

The semidirect product of a group N by a group H depends on a homomorphism Φ from H to the automorphism group of N.

There may be several such homomorphisms. If so, different homomorphisms will generate different semidirect products. The homomorphism mapping all H to the identity will generate the direct product of N and H.

Formally, we can regard each element of G as an ordered pair of elements of N,H. The group operation of G=N⋊H is ( na, hp ) * ( nb, hq ) = ( na * nb', hp * hq ) where nb' is what Φ(hp) maps nb to.

Intuitively, H acts on N by permuting its elements, but N does not affect H.

H is isomorphic to a subgroup of G, but not to a normal subgoup of G. The extension is said to be "split", meaning that as well as a homomorphism from G to H, there exists a homomorphism from H to G such that applying the latter then the former maps each element of H to itself.

• ## Central Extensions

If G is a central extension of N by H, then N is in the centre of G, i.e. all the elements of N commute with all the elements of G (and N is therefore Abelian). The extension is based on a map Ψ from H×H to N. If this map takes everything to the identity, we just get the direct product of N and H.

Formally, we can regard each element of G as an ordered pair of elements of N,H. The group operation of G=N↑H is ( na, hp ) * ( nb, hq ) = ( na * nb * Ψ(hp,hq),  hp * hq ). Ψ must be such that we end up with N↑H being associative.

Intuitively, H "sits on top of" N, and is not affected by N. H affects N by sometimes generating elements of N and raining them down into it. H is not a subgroup of G (though it may, coincidentally, be isomorphic to one).

A central extension can be regarded as a toll-bean extension.

• ## Mixed Extensions

A mixed extension conmbines the properties of a a semidirect product and a central extension. N is a normal subgroup but is not central. There is a homomorphism Φ from H to the automorphism group of N, and another map Ψ from H×H to N. The group operation of G=N⋊H is ( na, hp) * ( nb, hq ) = ( na * nb' * Ψ(hp,hq),   hp * hq ),   where nb' is what Φ(hp) maps nb to.

Here are some examples of all these four types of group extension