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group extension (Definition)

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:
$\displaystyle 1\longrightarrow G\longrightarrow E\longrightarrow H\longrightarrow 1.$
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
    1. $ H$ is isomorphic to a subgroup of $ E$, thus viewing $ H$ as a subgroup of $ E$,
    2. $ E=GH$, and
    3. $ G\cap H=\langle1\rangle$.
    Equivalently, $ E$ is a semidirect product of $ G$ and $ H$ if the short exact sequence
    $\displaystyle 1\longrightarrow G\longrightarrow E\stackrel{\alpha}{\longrightarrow} H\longrightarrow 1$
    splits. That is, there is a group homomorphism $ \phi\colon H\to E$ such that the composition
    $\displaystyle H\stackrel{\phi}{\longrightarrow}E\stackrel{\alpha}{\longrightarrow}H$
    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:
    $\displaystyle \begin{xy} *!C\xybox{ \xymatrix{1\ar@{=}[d]\ar[r]&G\ar@{=}[d]\ar[... ...ar@{=}[d] \ar[r]&1\ar@{=}[d]\\ 1\ar[r]&G\ar[r]&E_2\ar[r]&H\ar[r]&1.} } \end{xy}$
    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. If all the groups involved are abelian (only that $ E$ is abelian is necessary here), then we have an abelian extension.
    2. 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. If $ G$ is a cyclic group, then the extensions in question are called cyclic extensions.



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See Also: HNN extension

Also defines:  split extension, abelian extension, central extension, cyclic extension, extension problem
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Cross-references: cyclic group, center, necessary, abelian, even, equivalences, 5-lemma, commutative, equivalent, theory, bijection, map, isomorphism, identity map, composition, group homomorphism, subgroup, semidirect product, direct product, short exact sequence, well-defined, quotient group, normal subgroup, isomorphic, groups
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This is version 8 of group extension, born on 2005-07-20, modified 2005-08-16.
Object id is 7246, canonical name is GroupExtension.
Accessed 5870 times total.

Classification:
AMS MSC: 20J05 (Group theory and generalizations :: Connections with homological algebra and category theory :: Homological methods in group theory)
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