group extension
Let G and H be groups. A group E is called an extension
of G by H if
-
1.
G is isomorphic
to a normal subgroup
N of E, and
-
2.
H is isomorphic to the quotient group
E/N.
The definition is well-defined and it is convenient sometimes to
regard G as a normal subgroup of E. The definition can be
alternatively defined: E is an extension of G by H if there is
a short exact sequence of groups:
1⟶G⟶E⟶H⟶1. |
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 E an extension of H by G.
Remarks
-
•
Given any groups G and H, an extension of G by H exists: 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 E is 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
-
(a)
H is isomorphic to a subgroup
of E, thus viewing H as a subgroup of E,
-
(b)
E=GH, and
-
(c)
G∩H=⟨1⟩.
Equivalently, E is a semidirect product of G and H if the short exact sequence
1⟶G⟶Eα⟶H⟶1 splits. That is, there is a group homomorphism ϕ:H→E such that the composition
Hϕ⟶Eα⟶H 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↦hG.
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)↦gh 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 ghg-1=ˉh∈H. Then gh=ˉhg, so ghˉh-1=ˉhgˉh-1=ˉg∈G, or that hˉh-1=g-1ˉg. Therefore, h=ˉh.)
-
(a)
-
•
The extension problem in group theory is the classification of all extension groups of a given group G by a given group H. Specifically, it is a problem of finding all “inequivalent” extensions of G by H. Two extensions E1 and E2 of G by H are equivalent
if there is a homomorphism e:E1→E2 such that the following diagram of two short exact sequences is commutative
:
\xymatrix1\ar@=[d]\ar[r]&G\ar@=[d]\ar[r]&E1\ar[d]e\ar[r]&H\ar@=[d]\ar[r]&1\ar@=[d]1\ar[r]&G\ar[r]&E2\ar[r]&H\ar[r]&1. According to the 5-lemma, e is actually an isomorphism. Thus equivalences of extensions are well-defined.
-
•
Like split extensions, special extensions are formed when certain conditions are imposed on G, H, or even E:
-
(a)
If all the groups involved are abelian (only that E is abelian is necessary here), then we have an abelian extension
.
-
(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 (gˉh)h(gˉh)-1=ˉhhˉh-1∈H so that H is normal in E. Applying this result to the previous discussion and we have E≅G×H.
-
(c)
If G is a cyclic group
, then the extensions in question are called cyclic extensions.
-
(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 |