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:

• Direct product generally denoted by ×
• Semidirect product generally denoted by ⋊
• Central extension here denoted by ↑
• What I shall call mixed extension here denoted by ⋊↑

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 ( n_a, h_p ) * ( n_b, h_q ) = ( n_a*n_b, h_p*h_q ).

  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 (n_a, 1) commutes with every element of the form (1, h_p). 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 ( n_a, h_p ) * ( n_b, h_q ) = ( n_a * n_b', h_p * h_q ) where n_b' is what Φ(h_p) maps n_b 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 ( n_a, h_p ) * ( n_b, h_q ) = ( n_a * n_b * Ψ(h_p,h_q),  h_p * h_q ). 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 ( n_a, h_p) * ( n_b, h_q ) = ( n_a * n_b' * Ψ(h_p,h_q),   h_p * h_q ),   where n_b' is what Φ(h_p) maps n_b to.

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

See also my main index page for groups.

Copyright N.S.Wedd 2008