Extensions in General
One extension, the direct product, is immediately obvious. If one requires G and Q to be abelian groups, then the set of isomorphism classes of extensions of Q by a given (abelian) group N is in fact a group, which is isomorphic to
- ;
cf. the Ext 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 the wreath product provide further examples of extensions.
Read more about this topic: Group Extension
Famous quotes containing the words extensions and/or general:
“If we focus exclusively on teaching our children to read, write, spell, and count in their first years of life, we turn our homes into extensions of school and turn bringing up a child into an exercise in curriculum development. We should be parents first and teachers of academic skills second.”
—Neil Kurshan (20th century)
“Treating ‘water’ as a name of a single scattered object is not intended to enable us to dispense with general terms and plurality of reference. Scatter is in fact an inconsequential detail.”
—Willard Van Orman Quine (b. 1908)