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A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868501.png" /> generated by proper subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868502.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868503.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868504.png" /> normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868506.png" /> (so that the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868507.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868508.png" />, cf. [[Normal subgroup|Normal subgroup]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s0868509.png" /> is called a split extension of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685010.png" /> by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685011.png" />, or a semi-direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685013.png" />. If the subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685015.png" /> commute elementwise, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685018.png" />, their semi-direct product coincides with the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685019.png" />. A semi-direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685020.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685021.png" /> and a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685022.png" /> is given by a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685024.png" /> into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685025.png" /> of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685026.png" />. In this case, the formula
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$#A+1 = 47 n = 0
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$#C+1 = 47 : ~/encyclopedia/old_files/data/S086/S.0806850 Splittable group
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<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/s086/s086850/s08685027.png" /></td> </tr></table>
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{{TEX|done}}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685029.png" />, defines the multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685030.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685032.png" /> is the identity mapping, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685033.png" /> is called the holomorph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685034.png" /> (cf. [[Holomorph of a group|Holomorph of a group]]).
+
A [[Group|group]]  $  G $
 +
generated by proper subgroups  $  H $
 +
and  $  K $
 +
with  $  H $
 +
normal in  $  G $
 +
and  $  H \cap K = E $(
 +
so that the quotient group  $  G/H $
 +
is isomorphic to  $  K $,
 +
cf. [[Normal subgroup|Normal subgroup]]). $  G $
 +
is called a split extension of the group  $  H $
 +
by the group  $  K $,
 +
or a semi-direct product of  $  H $
 +
and  $  K $.  
 +
If the subgroups  $  H $
 +
and  $  K $
 +
commute elementwise, i.e.  $  hk = kh $
 +
for all  $  h \in H $,
 +
$  k \in K $,
 +
their semi-direct product coincides with the direct product  $  H \times K $.  
 +
A semi-direct product  $  G $
 +
of a group  $  H $
 +
and a group  $  K $
 +
is given by a homomorphism  $  \psi $
 +
of  $  K $
 +
into the group  $  \mathop{\rm Aut}  H $
 +
of automorphisms of  $  H $.  
 +
In this case, the formula
 +
 
 +
$$
 +
( h _ {1} , k _ {1} ) ( h _ {2} , k _ {2} )  = \
 +
( h _ {1} \psi ( k _ {1} ) ( h _ {2} ) , k _ {1} k _ {2} )
 +
$$
 +
 
 +
for all  $  h _ {1} , h _ {2} \in H $,
 +
$  k _ {1} , k _ {2} \in K $,  
 +
defines the multiplication in $  G $.  
 +
In the case when $  K = \mathop{\rm Aut}  H $
 +
and $  \psi $
 +
is the identity mapping, $  G $
 +
is called the holomorph of $  H $(
 +
cf. [[Holomorph of a group|Holomorph of a group]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Chelsea, reprint  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gorenstein,  "Finite groups" , Chelsea, reprint  (1980)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685035.png" /> is a semi-direct product, then [[conjugation]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685037.png" /> defines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685038.png" /> from which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685039.png" /> can be reconstructed, i.e.
+
Conversely, if $  G = HK $
 +
is a semi-direct product, then [[conjugation]] with $  k $
 +
in $  G $
 +
defines a homomorphism $  \psi : K \rightarrow  \mathop{\rm Aut}  H $
 +
from which $  G $
 +
can be reconstructed, i.e.
  
<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/s086/s086850/s08685040.png" /></td> </tr></table>
+
$$
 +
\psi ( k) ( h)  = k h k  ^ {-} 1 .
 +
$$
  
As a set the semi-direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685042.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685043.png" />. The subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685045.png" /> are subgroups that identify with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086850/s08685047.png" />.
+
As a set the semi-direct product of $  H $
 +
and $  K $
 +
is $  H \times K $.  
 +
The subsets $  \{ {( h , 1) } : {h \in H } \} $,
 +
$  \{ {( 1, k) } : {k \in K } \} $
 +
are subgroups that identify with $  H $
 +
and $  K $.

Revision as of 08:22, 6 June 2020


A group $ G $ generated by proper subgroups $ H $ and $ K $ with $ H $ normal in $ G $ and $ H \cap K = E $( so that the quotient group $ G/H $ is isomorphic to $ K $, cf. Normal subgroup). $ G $ is called a split extension of the group $ H $ by the group $ K $, or a semi-direct product of $ H $ and $ K $. If the subgroups $ H $ and $ K $ commute elementwise, i.e. $ hk = kh $ for all $ h \in H $, $ k \in K $, their semi-direct product coincides with the direct product $ H \times K $. A semi-direct product $ G $ of a group $ H $ and a group $ K $ is given by a homomorphism $ \psi $ of $ K $ into the group $ \mathop{\rm Aut} H $ of automorphisms of $ H $. In this case, the formula

$$ ( h _ {1} , k _ {1} ) ( h _ {2} , k _ {2} ) = \ ( h _ {1} \psi ( k _ {1} ) ( h _ {2} ) , k _ {1} k _ {2} ) $$

for all $ h _ {1} , h _ {2} \in H $, $ k _ {1} , k _ {2} \in K $, defines the multiplication in $ G $. In the case when $ K = \mathop{\rm Aut} H $ and $ \psi $ is the identity mapping, $ G $ is called the holomorph of $ H $( cf. Holomorph of a group).

References

[1] D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)

Comments

Conversely, if $ G = HK $ is a semi-direct product, then conjugation with $ k $ in $ G $ defines a homomorphism $ \psi : K \rightarrow \mathop{\rm Aut} H $ from which $ G $ can be reconstructed, i.e.

$$ \psi ( k) ( h) = k h k ^ {-} 1 . $$

As a set the semi-direct product of $ H $ and $ K $ is $ H \times K $. The subsets $ \{ {( h , 1) } : {h \in H } \} $, $ \{ {( 1, k) } : {k \in K } \} $ are subgroups that identify with $ H $ and $ K $.

How to Cite This Entry:
Splittable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splittable_group&oldid=48785
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article