Difference between revisions of "Splittable group"
From Encyclopedia of Mathematics
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| − | 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 <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. |
<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> | <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> | ||
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 <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" />. | ||
Revision as of 21:15, 29 November 2014
A group 
generated by proper subgroups 
and 
with 
normal in 
and 
(so that the quotient group 
is isomorphic to 
, cf. Normal subgroup). 
is called a split extension of the group 
by the group 
, or a semi-direct product of 
and 
. If the subgroups 
and 
commute elementwise, i.e. 
for all 
, 
, their semi-direct product coincides with the direct product 
. A semi-direct product 
of a group 
and a group 
is given by a homomorphism 
of 
into the group 
of automorphisms of 
. In this case, the formula
 |
for all 
, 
, defines the multiplication in 
. In the case when 
and 
is the identity mapping, 
is called the holomorph of 
(cf. Holomorph of a group).
References
| [1] | D. Gorenstein, "Finite groups" , Chelsea, reprint (1980) |
Comments
Conversely, if 
is a semi-direct product, then conjugation with 
in 
defines a homomorphism 
from which 
can be reconstructed, i.e.
 |
As a set the semi-direct product of 
and 
is 
. The subsets 
, 
are subgroups that identify with 
and 
.
How to Cite This Entry:
Splittable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splittable_group&oldid=35137
Splittable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splittable_group&oldid=35137
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