the kernel of a group homomorphism is a normal subgroup
In this entry we show the following simple lemma:
Let and be as in the statement of the lemma and let and . Then, by definition and:
where we have used several times the properties of group homomorphisms and the properties of the identity element . Thus, and is also an element of the kernel of . Since and were arbitrary, it follows that is normal in . ∎
Let be a group and let be a normal subgroup of . Then there exists a group homomorphism , for some group , such that the kernel of is precisely .
Although the first lemma is very simple, it is very useful when one tries to prove that a subgroup is normal.
Let be a field. Let us prove that the special linear group is normal inside the general linear group , for all . By the lemmas, it suffices to construct a homomorphism of with as kernel. The determinant of matrices is the homomorphism we are looking for. Indeed:
is a group homomorphism from to the multiplicative group and, by definition, the kernel is precisely , i.e. the matrices with determinant . Hence, is normal in .
|Title||the kernel of a group homomorphism is a normal subgroup|
|Date of creation||2013-03-22 17:20:34|
|Last modified on||2013-03-22 17:20:34|
|Last modified by||alozano (2414)|