Let and be groups. A group is called an extension of by if
is isomorphic to the quotient group .
The definition is well-defined and it is convenient sometimes to regard as a normal subgroup of . The definition can be alternatively defined: is an extension of by if there is a short exact sequence 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 an extension of by .
Given any groups and , an extension of by exists: take the direct product of and .
An intermediate concept between an extension a direct product is that of a semidirect product of two groups: If and are groups, and is an extension of by (identifying with a normal subgroup of ), then is called a semidirect product of by if
is isomorphic to a subgroup of , thus viewing as a subgroup of ,
Equivalently, is a semidirect product of and if the short exact sequence
Furthermore, if happens to be normal in , then is isomorphic to the direct product of and . (We need to show that 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 commutes with every element of . To show the last step, suppose . Then , so , or that . Therefore, .)
The extension problem in group theory is the classification of all extension groups of a given group by a given group . Specifically, it is a problem of finding all “inequivalent” extensions of by . Two extensions and of by are equivalent if there is a homomorphism such that the following diagram of two short exact sequences is commutative:
According to the 5-lemma, is actually an isomorphism. Thus equivalences of extensions are well-defined.
Like split extensions, special extensions are formed when certain conditions are imposed on , , or even :
If , considered as a normal subgroup of , actually lies within the center of , then is called a central extension. A central extension that is also a semidirect product is a direct product. Indeed, if is both a central extension and a semidirect product of by , we observe that so that is normal in . Applying this result to the previous discussion and we have .
|Date of creation||2013-03-22 15:24:25|
|Last modified on||2013-03-22 15:24:25|
|Last modified by||CWoo (3771)|