group extension
Let and be groups. A group is called an extension of by if
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1.
is isomorphic to a normal subgroup of , and
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2.
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 .
Remarks
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•
Given any groups and , an extension of by exists: take the direct product of and .
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•
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
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(a)
is isomorphic to a subgroup of , thus viewing as a subgroup of ,
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(b)
, and
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(c)
.
Equivalently, is a semidirect product of and if the short exact sequence
splits. That is, there is a group homomorphism such that the composition
gives the identity map. Thus, a semidirect product is also known as a split extension. That a semidirect product of by is also an extension of by can be seen via the isomorphism .
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, .)
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(a)
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•
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.
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•
Like split extensions, special extensions are formed when certain conditions are imposed on , , or even :
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(a)
If all the groups involved are abelian (only that is abelian is necessary here), then we have an abelian extension.
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(b)
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 .
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(c)
If is a cyclic group, then the extensions in question are called cyclic extensions.
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(a)
Title | group extension |
Canonical name | GroupExtension |
Date of creation | 2013-03-22 15:24:25 |
Last modified on | 2013-03-22 15:24:25 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 20J05 |
Related topic | HNNExtension |
Defines | split extension |
Defines | abelian extension |
Defines | central extension |
Defines | cyclic extension |
Defines | extension problem |