semi-direct factor and quotient group
Theorem.
If the group is a semi-direct product of its subgroups and ,
then the semi-direct
is isomorphic
to the quotient group
.
Proof. Every element of has the unique representation with and . We therefore can define the mapping
from to .
The mapping is surjective since any element of is the image of .
The mapping is also a homomorphism
since if and , then we obtain
Then we see that because all elements of
are mapped to the identity element of .
Consequently we get, according to the first isomorphism theorem
, the result
Example.
The multiplicative group of reals
is the semi-direct product of the subgroups
and .
The quotient group consists of all cosets
where , and is obviously isomorphic with .
Title | semi-direct factor and quotient group |
---|---|
Canonical name | SemidirectFactorAndQuotientGroup |
Date of creation | 2013-03-22 15:10:22 |
Last modified on | 2013-03-22 15:10:22 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 8 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 20E22 |