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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840701.png" /> by a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840702.png" />''
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''of a group $A$ by a group $B$''
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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 $a$:
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$$
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\alpha_a(b) = a b a^{-1}\,,\ \ \ b \in B \ .
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$$
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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.
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====References====
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<table>
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<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>
  
A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840703.png" /> which is the product of its subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840706.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840708.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s0840709.png" /> is also normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407010.png" />, then the semi-direct product becomes a [[Direct product|direct product]]. The semi-direct product of two groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407012.png" /> is not uniquely determined. To construct a semi-direct product one should also know which automorphisms of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407013.png" /> are induced by conjugation by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407014.png" />. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407015.png" /> is a semi-direct product, then to each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407016.png" /> corresponds an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407017.png" />, which is conjugation by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407018.png" />:
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407019.png" /></td> </tr></table>
 
  
Here, the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407020.png" /> is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407021.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407023.png" /> are arbitrary groups, then for any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407024.png" /> there is a unique semi-direct product of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407025.png" /> by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407027.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407028.png" />. A semi-direct product is a particular case of an extension of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407029.png" /> by a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407030.png" /> (cf. [[Extension of a group|Extension of a group]]); such an extension is called split.
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====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
 +
$$
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(a_1,b_1) \cdot (a_2,b_2) = \left({a_1a_2, b_1^{\phi(a_2)}b_2}\right) \ .
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$$
  
 
====References====
 
====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">[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>
  
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{{TEX|done}}
  
 
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[[Category:Group theory and generalizations]]
====Comments====
 
The semi-direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407031.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407032.png" /> is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407033.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084070/s08407034.png" />.
 

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=11367
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