group extension (original) (raw)

Remarks

• •
  Given any groups G and H, an extension of G by Hexists: take the direct product of G and H.
• •
  An intermediate concept between an extension a direct product is that of a semidirect product of two groups: If G and H are groups, and Eis 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 subgroup of E, thus viewing H as a subgroup of E,
  2. (b)
     E=G⁢H, and
  3. (c)
     G∩H=⟨1⟩.
     Equivalently, E is a semidirect product of G and H if the short exact sequence
     splits. That is, there is a group homomorphism ϕ:H→E 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 isomorphism h↦h⁢G.
     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)↦g⁢h is an isomorphism. It is not hard to see that the map is a bijection. The trick is to show that it is a homomorphism, which boils down to showing that every element of G commutes with every element of H. To show the last step, suppose g⁢h⁢g-1=h¯∈H. Then g⁢h=h¯⁢g, so g⁢h⁢h¯-1=h¯⁢g⁢h¯-1=g¯∈G, or that h⁢h¯-1=g-1⁢g¯. Therefore, h=h¯.)
• •
• •
  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 extension.
  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 (g⁢h¯)⁢h⁢(g⁢h¯)-1=h¯⁢h⁢h¯-1∈H so that H is normal in E. Applying this result to the previous discussion and we have E≅G×H.
  3. (c)