Let $G$ and $H$ be groups. A group $E$ is called an extension of $G$ by $H$ if

1. 1.

   $G$ is isomorphic to a normal subgroup $N$ of $E$, and

2. 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:

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

  1. (a)

     $H$ is isomorphic to a subgroup of $E$, thus viewing $H$ as a subgroup of $E$,

  2. (b)

     $E=GH$, and

  3. (c)

     $G\cap H=\langle 1\rangle$.

  Equivalently, $E$ is a semidirect product of $G$ and $H$ if the short exact sequence

  $$1\longrightarrow G\longrightarrow E\stackrel{\alpha}{\longrightarrow}H\longrightarrow 1$$

  splits. That is, there is a group homomorphism $\phi\colon H\to E$ such that the composition

  $$H\stackrel{\phi}{\longrightarrow}E\stackrel{\alpha}{\longrightarrow}H$$

  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 $ghg^{{-1}}=\overline{h}\in H$. 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, $h=\overline{h}$.)

• 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 $e\colon E_{1}\to E_{2}$ such that the following diagram of two short exact sequences is commutative:

  $$\xymatrix{1\ar@{=}[d]\ar[r]&G\ar@{=}[d]\ar[r]&E_{1}\ar[d]^{e}\ar[r]&H\ar@{=}[d]\ar[r]&1\ar@{=}[d]\\ 1\ar[r]&G\ar[r]&E_{2}\ar[r]&H\ar[r]&1.}$$

  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 $E$:

  1. (a)

     If all the groups involved are abelian (only that $E$ is abelian is necessary here), then we have an abelian extension.

  2. (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 $E\cong G\times H$.

  3. (c)

     If $G$ is a cyclic group, then the extensions in question are called cyclic extensions.