Semi-direct product
of a group by a group
A group which is the product of its subgroups
and
, where
is normal in
and
. If
is also normal in
, then the semi-direct product becomes a direct product. The semi-direct product of two groups
and
is not uniquely determined. To construct a semi-direct product one should also know which automorphisms of the group
are induced by conjugation by elements of
. More precisely, if
is a semi-direct product, then to each element
corresponds an automorphism
, which is conjugation by the element
:
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Here, the correspondence is a homomorphism
. Conversely, if
and
are arbitrary groups, then for any homomorphism
there is a unique semi-direct product of the group
by the group
for which
for any
. A semi-direct product is a particular case of an extension of a group
by a group
(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 by
is often denoted by
or
.
Semi-direct product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-direct_product&oldid=11367