Difference between revisions of "Splittable group"
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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
.
Splittable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splittable_group&oldid=14172