Difference between revisions of "Semi-direct product"
(Category:Group theory and generalizations) |
(Terminology: internal and external, cite Cohn (2003)) |
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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 [[ | + | 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. [[ | + | 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> | + | <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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====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 | ||
+ | $$ | ||
+ | (a_1,b_1) \cdot (a_2,b_2) = \left({a_1a_2, b_1^{\phi(a_2)}b_2}\right) \ . | ||
+ | $$ | ||
+ | |||
+ | ====References==== | ||
+ | <table> | ||
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> Paul M. Cohn. ''Basic Algebra: Groups, Rings, and Fields'', Springer (2003) ISBN 1852335874. Zbl 1003.00001</TD></TR> | ||
+ | </table> | ||
{{TEX|done}} | {{TEX|done}} | ||
[[Category:Group theory and generalizations]] | [[Category:Group theory and generalizations]] |
Revision as of 16:16, 19 October 2014
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
[1] | Paul M. Cohn. Basic Algebra: Groups, Rings, and Fields, Springer (2003) ISBN 1852335874. Zbl 1003.00001 |
Semi-direct product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-direct_product&oldid=33934