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Difference between revisions of "Semi-direct product"

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''of a group $A$ by a group $B$''
 
''of a group $A$ by a group $B$''
  
A group $G = AB$ which is the product of its subgroups $A$ and $B$, where $B$ is normal in $G$ and $A \cap B = \{1\}$. If $A$ is also normal in $G$, then the semi-direct product becomes a [[Direct product|direct product]]. The semi-direct product of two groups $A$ and $B$ is not uniquely determined. To construct a semi-direct product one should also know which automorphisms of the group $B$ are induced by conjugation by elements of $A$. More precisely, if $G = AB$ is a semi-direct product, then to each element $a \in A$ corresponds an automorphism $\alpha_a \in \mathrm{Aut}(B)$, which is conjugation by the element $a$:
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A group $G = AB$ which is the product of its subgroups $A$ and $B$, where $B$ is normal in $G$ and $A \cap B = \{1\}$. If $A$ is also normal in $G$, then the semi-direct product becomes a [[direct product]]. The semi-direct product of two groups $A$ and $B$ is not uniquely determined. To construct a semi-direct product one should also know which automorphisms of the group $B$ are induced by conjugation by elements of $A$. More precisely, if $G = AB$ is a semi-direct product, then to each element $a \in A$ corresponds an automorphism $\alpha_a \in \mathrm{Aut}(B)$, which is [[conjugation]] by the element $a$:
 
$$
 
$$
 
\alpha_a(b) = a b a^{-1}\,,\ \ \ b \in B \ .
 
\alpha_a(b) = a b a^{-1}\,,\ \ \ b \in B \ .
 
$$
 
$$
Here, the correspondence $a \mapsto \alpha_a$ is a homomorphism $A \rightarrow \mathrm{Aut}(B)$. Conversely, if $A$ and $B$  are arbitrary groups, then for any homomorphism $\phi  : A \rightarrow \mathrm{Aut}(B)$ there is a unique semi-direct product of the group $A$ by the group $B$ for which $\alpha_a = \phi(a)$ for any $a \in A$. A semi-direct product is a particular case of an extension of a group $B$ by a group $A$ (cf. [[Extension of a group|Extension of a group]]); such an extension is called split.
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Here, the correspondence $a \mapsto \alpha_a$ is a homomorphism $A \rightarrow \mathrm{Aut}(B)$. Conversely, if $A$ and $B$  are arbitrary groups, then for any homomorphism $\phi  : A \rightarrow \mathrm{Aut}(B)$ there is a unique semi-direct product of the group $A$ by the group $B$ for which $\alpha_a = \phi(a)$ for any $a \in A$. A semi-direct product is a particular case of an extension of a group $B$ by a group $A$ (cf. [[Extension of a group]]); such an extension is called split.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1''' , Chelsea  (1960)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1''' , Chelsea  (1960)  (Translated from Russian)</TD></TR>
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</table>
  
  
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====Comments====
 
====Comments====
 
The semi-direct product of $A$ by $B$ is often denoted by $B \rtimes A$ or $B : A$.
 
The semi-direct product of $A$ by $B$ is often denoted by $B \rtimes A$ or $B : A$.
 +
 +
The term "internal" semi-direct product is used for the case when $A$ and $B$ are considered as subgroups of the given group $G$.  The "external" semi-direct product of groups $A$ and $B$, with a map $\phi  : A \rightarrow \mathrm{Aut}(B)$ , may be taken to be the [[Cartesian product]] $A \times B$ with multiplication defined by
 +
$$
 +
(a_1,b_1) \cdot (a_2,b_2) = \left({a_1a_2, b_1^{\phi(a_2)}b_2}\right) \ .
 +
$$
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====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  Paul M. Cohn.  ''Basic Algebra: Groups, Rings, and Fields'', Springer (2003) {{ISBN|1852335874}} {{ZBL|1003.00001}}</TD></TR>
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</table>
  
 
{{TEX|done}}
 
{{TEX|done}}
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[[Category:Group theory and generalizations]]

Latest revision as of 16:54, 23 November 2023

of a group $A$ by a group $B$

A group $G = AB$ which is the product of its subgroups $A$ and $B$, where $B$ is normal in $G$ and $A \cap B = \{1\}$. If $A$ is also normal in $G$, then the semi-direct product becomes a direct product. The semi-direct product of two groups $A$ and $B$ is not uniquely determined. To construct a semi-direct product one should also know which automorphisms of the group $B$ are induced by conjugation by elements of $A$. More precisely, if $G = AB$ is a semi-direct product, then to each element $a \in A$ corresponds an automorphism $\alpha_a \in \mathrm{Aut}(B)$, which is conjugation by the element $a$: $$ \alpha_a(b) = a b a^{-1}\,,\ \ \ b \in B \ . $$ Here, the correspondence $a \mapsto \alpha_a$ is a homomorphism $A \rightarrow \mathrm{Aut}(B)$. Conversely, if $A$ and $B$ are arbitrary groups, then for any homomorphism $\phi : A \rightarrow \mathrm{Aut}(B)$ there is a unique semi-direct product of the group $A$ by the group $B$ for which $\alpha_a = \phi(a)$ for any $a \in A$. A semi-direct product is a particular case of an extension of a group $B$ by a group $A$ (cf. Extension of a group); such an extension is called split.

References

[1] A.G. Kurosh, "The theory of groups" , 1 , Chelsea (1960) (Translated from Russian)


Comments

The semi-direct product of $A$ by $B$ is often denoted by $B \rtimes A$ or $B : A$.

The term "internal" semi-direct product is used for the case when $A$ and $B$ are considered as subgroups of the given group $G$. The "external" semi-direct product of groups $A$ and $B$, with a map $\phi : A \rightarrow \mathrm{Aut}(B)$ , may be taken to be the Cartesian product $A \times B$ with multiplication defined by $$ (a_1,b_1) \cdot (a_2,b_2) = \left({a_1a_2, b_1^{\phi(a_2)}b_2}\right) \ . $$

References

[a1] Paul M. Cohn. Basic Algebra: Groups, Rings, and Fields, Springer (2003) ISBN 1852335874 Zbl 1003.00001
How to Cite This Entry:
Semi-direct product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-direct_product&oldid=33932
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article