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 welldefined 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:
$$1\u27f6G\u27f6E\u27f6H\u27f61.$$ 
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)
$G\cap H=\u27e81\u27e9$.
Equivalently, $E$ is a semidirect product of $G$ and $H$ if the short exact sequence
$$1\u27f6G\u27f6E\stackrel{\alpha}{\u27f6}H\u27f61$$ splits. That is, there is a group homomorphism $\varphi :H\to E$ such that the composition
$$H\stackrel{\varphi}{\u27f6}E\stackrel{\alpha}{\u27f6}H$$ 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^{} $h\mapsto hG$.
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 $(g,h)\mapsto gh$ 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 $gh{g}^{1}=\overline{h}\in H$. Then $gh=\overline{h}g$, so $gh{\overline{h}}^{1}=\overline{h}g{\overline{h}}^{1}=\overline{g}\in G$, or that $h{\overline{h}}^{1}={g}^{1}\overline{g}$. Therefore, $h=\overline{h}$.)

(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 ${E}_{1}$ and ${E}_{2}$ of $G$ by $H$ are equivalent^{} if there is a homomorphism $e:{E}_{1}\to {E}_{2}$ such that the following diagram of two short exact sequences is commutative^{}:
$$\text{xymatrix}1\text{ar}\mathrm{@}=[d]\text{ar}[r]\mathrm{\&}G\text{ar}\mathrm{@}=[d]\text{ar}[r]\mathrm{\&}{E}_{1}\text{ar}{[d]}^{e}\text{ar}[r]\mathrm{\&}H\text{ar}\mathrm{@}=[d]\text{ar}[r]\mathrm{\&}1\text{ar}\mathrm{@}=[d]1\text{ar}[r]\mathrm{\&}G\text{ar}[r]\mathrm{\&}{E}_{2}\text{ar}[r]\mathrm{\&}H\text{ar}[r]\mathrm{\&}1.$$ According to the 5lemma, $e$ is actually an isomorphism. Thus equivalences of extensions are welldefined.

•
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 $(g\overline{h})h{(g\overline{h})}^{1}=\overline{h}h{\overline{h}}^{1}\in H$ so that $H$ is normal in $E$. Applying this result to the previous discussion and we have $E\cong G\times H$.

(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  20130322 15:24:25 
Last modified on  20130322 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 